On the Balance between Emigration and Immigration as Random Walks on Non-Negative Integers

Abstract

Life is on a razor’s edge resulting from the random competitive forces of birth and death. We illustrate this aphorism in the context of three Markov chain population models where systematic random immigration events promoting growth are simultaneously balanced with random emigration ones provoking thinning. The origin of mass removals is either determined by external demands or by aging, leading to different conditions of stability.

1. Introduction

Catastrophic events striking some population can cause a major death toll, and they hopefully occur rarely at hectic times. In calm times, the population can simply grow safely, maybe at a random pace. Catastrophe models are based on this idea that birth and death are exclusive events. The binomial catastrophe model is when, during a catastrophic event, the individuals of the current population each can die or survive in an independent and even way with some probability, resulting in a drastic depletion of individuals at each catastrophic step. For such systems, there is then a competition between random growth and declining forces, resulting in a subtle balance of the two. They can be handled in the context of discrete-time Markov chains on non-negative integers; see [1,2,3,4].

In [1], a catastrophe random walk model was also introduced in which the origin of the removal of individuals was based on a “truncated geometric” model. In other words, if a geometric catastrophic event occurs, given the population is in some state, its size further shrinks by a random (geometrically distributed) amount so long as this amount does not exceed the current state; if it does, the population size is set to $\left\{0\right\}$, a disastrous event [5]. So, the random sequential thinning of the population keeps going on but is stopped as soon as the current population size is exhausted. In calm times, the population is incremented by a random amount. The geometric effect corresponds to softer depletion issues than for the binomial model. This random walk may also be viewed as giving the size of some population facing (possibly) accidentally a steady random geometric demand of emigrants from outside or alternatively being revitalized by a steady number of immigrants. This model was recently further analyzed in [6]. More examples and motivations can be found in [7,8] (essentially in continuous times).

Putting aside the catastrophe idea, one can think of a similar random walk process now giving the size of some population facing systematically a steady random demand of emigrants from outside and simultaneously being revitalized by a steady number of immigrants. Alternatively, this Markov chain is a model for the state of some stock resulting from the competition between successive random supplies systematically balanced by simultaneous truncated geometric random demands. This results in a Markov chain of a new type.

All such Markov chain models have been designed in an attempt to explain the transient and large time behavior of the population size. Some results concern the evaluation of the risk of extinction and the distribution of the population size in the case of total disasters where all individuals in the population are removed simultaneously [5,9]. Such Markov chains are random walks on non-negative integers (as a semigroup), which differ from standard random walks on integers (as a group) in that a one-step move down from some positive integer cannot take the walker to a negative state, resulting in transition probabilities being state-dependent.

In Section 2 of this work, we first study the variation in the Neuts’ truncated geometric catastrophe model in discrete time, in which input and output can occur simultaneously. It is also a Markov chain on non-negative integers.

Using a generating function approach, we first discuss the condition under which this process is recurrent (either positive or null) or transient. The recurrence/transience phase transition is sharp and it easily occurs due to moderate depletion in the shrinkage steps of this model.

To this end, a recurrence relation for the probability-generating function of this process is first derived (Proposition 1). It is used to discriminate the phase transition between recurrent and transient regimes.

In the positive-recurrent (subcritical) regime, we describe the shape and features of the invariant probability measure (Theorem 1).We emphasize that in the null-recurrent (critical) regime, no non-trivial invariant measure exists. We then show how to compute the double (space/time) generating functional of the process (Proposition 2). Using this representation, we derive the generating functions of the first return time to zero (the length of the excursions) and of the first local extinction time (else first hitting time of 0) when the process is started at $x_0>0$. In the recurrent regime, a first local extinction occurs with probability 1. The analyses rely on the existence of a ‘key’ function which is the inverse of the singularity of the double generating function of the process. The Lagrange inversion formula yields a power-series expansion of the ‘key’ function (Proposition 3). It is used to prove a decomposition result for the first local extinction time (Theorem 2).

In the transient (supercritical) regime, after a finite number of visits to $\left\{0\right\}$, the chain drifts to ∞. We first emphasize that no non-trivial invariant measure exists either. Using the expression of the double generating functional of the process, we obtain access to a precise large deviation result (Proposition 4). The harmonic (or scale) function of a version of the process forced to be absorbed in state $\left\{0\right\}$ is then used to give an expression of the probability that extinction occurs before the explosion (Proposition 5).

In Section 3, two related emigration/immigration models are analyzed.

One is a model related to the truncated geometric emigration/immigration one in Section 2; it was recently introduced in [10]. In one of its interpretations, the state variable is now the number of individuals that could potentially be released in the latter truncated geometric model: the chain’s state is the system’s capacity of release. This ‘dual’ chain is analyzed along similar lines as before and it is shown to exhibit very different statistical features; in particular it is emphasized that the corresponding Markov chain is always positive-recurrent (subcritical) whatever the law of the input (Proposition 6), a remarkable stability property. Using the expression of its double generating function (Proposition 7), we further show that, in this context, the first hitting time of 0 when starting from $x_0\ge 1$ is independent of $x_0$ and geometrically distributed (Corollary 1).

For comparison, we also briefly revisit the subcritical Bienaymé–Galton–Watson branching process with immigration, for which the origin of the pruning of individuals is rather internal and due to the intrinsic imbalance of birth and death events inside the population. In such models, both emigration and immigration also can occur simultaneously as well, and so they do not have the flavor of catastrophe either. The conditions for the recurrence/transience transition are recalled and detailed; they are weak due to massive depletion of individuals in the shrinkage steps. In contrast with the truncated geometric model, the critical regime for the underlying branching process exhibits a non-trivial asymptotic behavior, following Seneta’s results [11].

All models involve two stationary stochastic sources, one determining the thinning of the population and the other competing one, which is additive, its growth. In this sense, they are bi-stochastic. Semi-stochastic growth/collapse or decay/surge Piecewise Deterministic Markov Processes are in the same vein, although in the continuum, they were recently considered in [12,13], following the work [14], where physical applications were developed, such as queueing processes arising in the physics of dams or stress release issues arising in the physics of earthquakes. Here, growth was determined by a deterministic differential equation and the collapse effect was random and state-dependent.

2. Truncated Geometric Models

In [6], the following random walk on the non-negative integers was revisited, following the work of [1]. It is a Markov chain modeling the competition between successive random supplies ${\beta }_n$ balanced by truncated geometric random demands ${\delta }_n$. The random walk process gives the size $X_n$ of some population facing accidentally a steady random demand ${\delta }_n$ of emigrants from outside or alternatively being revitalized by a steady number ${\beta }_n$ of immigrants. The birth and death sequences were as follows:

• Birth (growth, immigration):

The sequence ${\left({\beta }_n\right)}_{n\ge 1}$ was an independent and identically distributed (i.i.d.) sequence taking values in $\mathbb{N}:=\left\{1,2,...\right\},$ with common law $b_x:=$$\mathbf{P}$$\left(\beta =x\right)$, $x\ge 1$, obeying $b_0=0$.

• Death (thinning, emigration):

The sequence ${\left({\delta }_n\right)}_{n\ge 1}$ was an i.i.d. shifted geometric $\left(\alpha \right)—$distributed one (A geometric $\left(\alpha \right)$ rv with success probability $\alpha$ takes values in $\mathbb{N}$. a shifted geometric $\left(\alpha \right)$ rv with success probability $\alpha$ takes values in ${\mathbb{N}}_0$; it is obtained while shifting the former one by one unit), with failure parameter $\alpha \in \left(0,1\right)$, viz. $\mathbf{P}\left(\delta =x\right)=\alpha {\overline{\alpha }}^x$, $x\ge 0$ (where $\overline{\alpha }:=1-\alpha$). Due to the memory-less property of $\delta$, the parameter $\alpha$ is alternatively the (constant) discrete failure rate of $\delta ,$ namely $\alpha =\mathbf{P}\left(\delta =x\right)/\mathbf{P}\left(\delta \ge x\right)$, $x\ge 0.$

The Markov chain under study was given by [1]:

(1)$X_{n+1}=\left\{\begin{array}{c}{X}_n+{\beta }_{n+1}\mathrm{with}\mathrm{probability}p\\ {\left(X_n-{\delta }_{n+1}\right)}_{+}\mathrm{with}\mathrm{probability}q=1-p,\end{array}\right.$

with alternating, so they are exclusive, immigration, and emigration events. The sequence ${\beta }_n$ was assumed to be independent of ${\delta }_n$ and $X_{n-k}$ for all $k\ge 1.$ The truncated emigration component in (1) is a non-linear random function of the state $X_n$.

At each step n, the walker either moves up with probability p, the amplitude of the upward move being ${\beta }_{n+1}$ or, given the chain is in state x, the number of step-wise removed individuals is ${\delta }_{n+1}\wedge x:=min\left({\delta }_{n+1},x\right),$ with probability q.

Note that if $X_n=0$, then $X_{n+1}={\beta }_{n+1}$ with probability p (reflection at 0) and $X_{n+1}=0$ with probability q (absorption at 0).

Remark 1.

Considerable simplifications are expected when $\beta =1$ with probability 1: in this case, the transition matrix P associated with (1) is of the Hessenberg type, which is lower triangular with a non-zero upper diagonal. The corresponding process is a skip-free to the right Markov chain. If, in addition, $\delta =1$ with probability 1 (which is ruled out here because δ was assumed here geometrically distributed), we are left with the standard birth and death chain with a tridiagonal transition matrix for which a non-trivial extension would be to have the probabilities p and q to move up and down depending on the current state $x$ [15]. Conditions of ergodicity of such chains are well known and have been studied for a long time; see [16], for example. The Neuts’ random walk is thus a generalized birth and death one with state-dependent probability transitions. Exclusivity of birth and death events suggest that in a typical regime, births are at stake, whereas, possibly exceptionally if q is small, a catastrophic event with a death toll occurs [6,15]. In the following version of this model, there is no longer such an interpretation of rare catastrophic events preventing growth.

2.1. Simultaneous Growth and Depletion: A New Truncated Geometric Model

We proceed with a similar study of the following modified version of the Neuts’ process, deserving special study and with specific statistical features. Consider the time-homogeneous Markov chain now with temporal evolution:

(2)$X_{n+1}={\left(X_n-{\delta }_{n+1}\right)}_{+}+{\beta }_{n+1};X_0\ge 0,\mathrm{else}$

$\mathsf{\Delta }{X}_n=X_{n+1}-X_n=-X_n\wedge {\delta }_{n+1}+{\beta }_{n+1}$

One of its possible observables is

$Y_{n+1}=-X_n\wedge {\delta }_{n+1},$

where the number of individuals currently effectively moves out the system. Based on observed $\left(y_n\right)$, de-trending can be used to estimate $\left(x_n\right).$

In contrast with the previous model (1), competing depletion and growth mechanisms can occur simultaneously; the two are no longer exclusive. As before, $X_n$ may represent the amount of some resource available at time n or the state of the fortune of some investor facing recurrent expenses but sustained by recurrent income. Due to the demand ${\delta }_{n+1}$ on day $n+1$, the stock shrinks by the random amount $S\left(X_n\right):=X_n\wedge {\delta }_{n+1}$, and, concomitantly, there is a simultaneous production ${\beta }_{n+1}$ of this resource used to face forthcoming demands. One can also loosely think of $X_n$ as a model of the height of a random polymer in a stationary random environment (see [17] and references therein). Polymer models as ‘classical’ space-inhomogeneous birth and death Markov chains on ${\mathbb{N}}_0:=\left\{0,1,2,...\right\}$ (with moves up and down only by one unit) were considered in [18], for example. They exhibit a pinning/depinning transition.

The birth and death sequences of our model are now chosen as follows:

• Growth [input]:

${\left({\beta }_n\right)}_{n\ge 1}$ is an i.i.d. sequence now taking values in ${\mathbb{N}}_0,$ with common law $b_x:=\mathbf{P}\left(\beta =x\right)$, $x\ge 0$. We assume $0<b_0<1.$ We shall let

${\varphi }_{\beta }\left(z\right):=\mathbf{E}\left(z^{\beta }\right)$

be the common probability generating function (p.g.f.) of the $\beta$s. There is now a positive probability $b_0$ that $\beta =0.$

• Depletion [output] (thinning):

${\left({\delta }_n\right)}_{n\ge 1}$ is an i.i.d. shifted geometric $\left(\alpha \right)—$distributed sequence (independent of ${\left({\beta }_n\right)}_{n\ge 1}$), with failure parameter $\alpha \in \left(0,1\right)$. Each $\delta$ then takes values in ${\mathbb{N}}_0$ with common law $\mathbf{P}\left(\delta =x\right)=:d_x=\alpha {\overline{\alpha }}^x$, $x\ge 0$ and mean ${\mu }_{\delta }:=\mathbf{E}\left(\delta \right)=\overline{\alpha }/\alpha$. State 0 is not absorbing (else reflecting) because $\mathbf{P}\left(X_1=0\mid X_0=0\right)=b_0<1:$ after some geometric random number of steps, the walker is bounced back inside the state–space. There is a positive probability $d_0=\alpha$ that $\delta =0$.

The sequence ${\beta }_n$ is assumed to be independent of ${\delta }_n$ and $X_{n-k}$ for all $k\ge 1.$

The one-step stochastic transition matrix P (obeying $P\mathbf{1}=\mathbf{1}$, where $\mathbf{1}$ is a column vector of ones) of the Markov chain $X_n$ is (with $d_z=0$ if $z<0$ and the convention ${\sum }_{z=a}^b=0$ if $b<a$):

(3)$P\left(x,y\right)=\sum _{z=x-y}^{x-1}{b}_{y-x+z}{d}_z+\mathbf{P}\left(\delta \ge x\right)b_y,x,y\ge 0,$

Note $P\left(0,y\right)=b_y$, $y\ge 0$ and $P\left(x,0\right)=\mathbf{P}\left(\delta \ge x\right)b_0$. In particular also (if emigration equals immigration), $P\left(x,x\right)={\sum }_{z=0}^{x-1}{b}_z{d}_z+\mathbf{P}\left(\delta \ge x\right)b_x$ are its diagonal matrix elements. This Markov chain is time-homogeneous, irreducible, and aperiodic if and only if $0<b_0<1$, with state $\left\{0\right\}$ reflecting. As a result, all states $x=\left\{0,1,2,...\right\}$ are either recurrent or transient. Note that the likelihood of a path $\left(x_0,...,x_n\right)$

$\prod _{m=1}^{n}P\left(x_{m-1},x_m\right)$

has a very complex structure.

Remark 2.

(2) suggests the continuous-time version of this counting process:

$d{X}_t=-X_{t_{-}}\wedge {\delta }_t·dt+d{N}_t$

where ${\delta }_t\stackrel{d}{\sim }\delta$ for all $t\ge 0$ is a strongly stationary geometric process and $N_t$ a compound Poisson process with discrete jumps’ amplitudes $\beta .$ We shall not run here into the analysis of this process, postponing it to future work.

A continuous space–time version of this process with state–space $\left[0,\infty \right)$ would be when ${\delta }_t\stackrel{d}{\sim }\delta$ is a strongly stationary process with δ exponentially distributed and when allowing β now to take values in the half-line or, more generally, when allowing $N_t$ to be a general Lévy subordinator.

2.2. The Recurrence Relation for the p.g.f. of $X_n$

For any given $X_n=x,$ the shrinkage random variable (r.v.): $S\left(x\right):={\left(x-\delta \right)}_{+}$ obeys $S\left(x\right)\le x$, having support on $\left\{0,...,x\right\}$. Hence, $S\left(X_n\right)\le X_n$ almost surely (a.s.). With ${\epsilon }_x$ the Dirac mass at x, the law of $S\left(x\right)$ is obtained as follows:

$\begin{array}{ccc}& & S\left(x\right)\stackrel{d}{\sim }{\overline{\alpha }}^x{\epsilon }_0+\sum _{y=0}^{x-1}\alpha {\overline{\alpha }}^y{\epsilon }_{x-y}\hfill \\ \hfill \mathbf{E}\left(z^{S\left(x\right)}\right)& =& {\overline{\alpha }}^x+\alpha \sum _{y=0}^{x-1}{\overline{\alpha }}^y{z}^{x-y}={\overline{\alpha }}^x+\alpha {\overline{\alpha }}^{x}z{\displaystyle \frac{1-{\left(z/\overline{\alpha }\right)}^x}{\overline{\alpha }-z}}\hfill \end{array}$

leading to the local drift and variance terms

$\begin{array}{ccc}\hfill f\left(x\right)& :& =\mathbf{E}\left(X_{n+1}-x\mid X_n=x\right)=\mathbf{E}\left(S\left(x\right)\right)-x+\mathbf{E}\left(\beta \right)=-{\displaystyle \frac{\overline{\alpha }}{\alpha }}\left(1-{\overline{\alpha }}^x\right)+\mathbf{E}\left(\beta \right)\hfill \\ & \sim & -{\displaystyle \frac{\overline{\alpha }}{\alpha }}+\mathbf{E}\left(\beta \right)\mathrm{as}x\to \infty .\hfill \\ \hfill {\sigma }^2\left(x\right)& :& ={\sigma }^2\left(\left(X_{n+1}-x\right)\mid X_n=x\right)={\sigma }^2\left(S\left(x\right)\right)+{\sigma }^2\left(\beta \right)\hfill \\ & =& {\displaystyle \frac{\overline{\alpha }}{{\alpha }^2}}-{\displaystyle \frac{2x+1}{\alpha }}{\overline{\alpha }}^{x+1}-{\left({\displaystyle \frac{{\overline{\alpha }}^{x+1}}{\alpha }}\right)}^2+{\sigma }^2\left(\beta \right)\hfill \\ & \sim & {\sigma }^2\left(\delta \right)+{\sigma }^2\left(\beta \right)\mathrm{as}x\to \infty .\hfill \end{array}$

If ${\mu }_{\beta }:=\mathbf{E}\left(\beta \right)<\infty$ and x is large, the walker ‘feels’ the constant drift $f{\stackrel{·}{\left(x\right)}}_{x\to \infty }-{\displaystyle \frac{\overline{\alpha }}{\alpha }}+\mathbf{E}\left(\beta \right)={\mu }_{\beta }-{\mu }_{\delta }$, positive or negative, depending on ${\mu }_{\beta }\gtrless {\mu }_{\delta }.$ Fluctuations are of order ${\sigma }^2\left(\delta \right)+{\sigma }^2\left(\beta \right)$ for large x. Averaging over $X_n$, we obtain

Proposition 1.

(4) ${\mathrm{\Phi }}_{n+1}\left(z\right):=\mathbf{E}\left(z^{X_{n+1}}\right)={\varphi }_{\beta }\left(z\right)\sum _{x\ge 0}\mathbf{P}\left(X_n=x\right)\left({\overline{\alpha }}^x+\alpha {\overline{\alpha }}^{x}z{\displaystyle \frac{1-{\left(z/\overline{\alpha }\right)}^x}{\overline{\alpha }-z}}\right)$

$\begin{array}{ccc}& =& {\varphi }_{\beta }\left(z\right)\left({\mathrm{\Phi }}_n\left(\overline{\alpha }\right)+{\displaystyle \frac{\alpha z}{\overline{\alpha }-z}}\left({\mathrm{\Phi }}_n\left(\overline{\alpha }\right)-{\mathrm{\Phi }}_n\left(z\right)\right)\right)\hfill \\ & =& {\varphi }_{\beta }\left(z\right)\left({\displaystyle \frac{\overline{\alpha }\left(1-z\right)}{\overline{\alpha }-z}}{\mathrm{\Phi }}_n\left(\overline{\alpha }\right)-{\displaystyle \frac{\alpha z}{\overline{\alpha }-z}}{\mathrm{\Phi }}_n\left(z\right)\right)\hfill \end{array}$

With ${\mathbf{ß}}_n:={\left({\mathbf{P}}_{x_0}\left(X_n=0\right),{\mathbf{P}}_{x_0}\left(X_n=1\right),...\right)}^{\prime }$, the column vector (in the sequel, a boldface variable, say $\mathbf{x}$, will represent a column vector so that its transpose, say ${\mathbf{x}}^{\prime }$, will be a row vector) of the states’ occupation probabilities at time n, ${\mathbf{ß}}_{n+1}^{\prime }={\mathbf{ß}}_n^{\prime }P$, $X_0\stackrel{d}{\sim }{\delta }_{x_0},$ is the master equation of its temporal evolution and (4) is the corresponding temporal evolution of ${\mathrm{\Phi }}_n\left(z\right)={\sum }_{x\ge 0}{z}^x{\mathbf{P}}_{x_0}\left(X_n=x\right).$

2.3. Positive Recurrence and the Invariant Probability Measure (Subcritical Regime)

If ${\mathrm{\Phi }}_{\infty }\left(z\right)$ were to exist, it should solve ${\mathrm{\Phi }}_{\infty }\left(z\right)={\varphi }_{\beta }\left(z\right)\left({\displaystyle \frac{\overline{\alpha }\left(1-z\right)}{\overline{\alpha }-z}}{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)-{\displaystyle \frac{\alpha z}{\overline{\alpha }-z}}{\mathrm{\Phi }}_{\infty }\left(z\right)\right)$ which is

(5)${\mathrm{\Phi }}_{\infty }\left(z\right)={\displaystyle \frac{\overline{\alpha }{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right){\varphi }_{\beta }\left(z\right)\left(1-z\right)}{1-z-\alpha \left(1-z{\varphi }_{\beta }\left(z\right)\right)}}={\displaystyle \frac{\mathcal{N}\left(z\right)}{\mathcal{D}\left(z\right)}}.$

Note that ${\mathrm{\Phi }}_{\infty }\left(0\right)=b_0{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)$ would be the probability that the chain is asymptotically in state $\left\{0\right\}.$ The solution ${\mathrm{\Phi }}_{\infty }\left(0\right)={\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)=0$ (suggesting transience) is incompatible with recurrence and the fact that state $\left\{0\right\}$ is non-absorbing. The chain (2), being irreducible and aperiodic, is sufficient for its positive recurrence to show that ${\mathrm{\Phi }}_n\left(0\right)$ approaches a positive limit as $n\to \infty$, since ${\mathrm{\Phi }}_n\left(0\right)=\mathbf{P}\left(X_n=0\right)=P^n\left(x_0,0\right)$ is the n-step transition probability from state $x_0$ to state $\left\{0\right\}$ of the chain.

Suppose $X_{\infty }\stackrel{d}{\sim }C{\delta }_0+\left(1-C\right){\delta }_{\infty }$ for some $C\in \left(0,1\right).$ Then ${\mathrm{\Phi }}_{\infty }\left(z\right)=C$ for all $z\in \left[0,1\right]$ and

$1={\displaystyle \frac{{\varphi }_{\beta }\left(z\right)\left(1-z\right)}{1-z-\alpha \left(1-{\varphi }_{\beta }\left(z\right)\right)}}$

which cannot be true unless ${\varphi }_{\beta }\left(z\right)=1$ ($\beta \stackrel{d}{\sim }{\delta }_0$), which is ruled out. However, ${\mathrm{\Phi }}_{\infty }\left(z\right)\equiv 0$ for all $z\in \left[0,1\right]$ ($C=0$ and $X_{\infty }\stackrel{d}{\sim }{\delta }_{\infty }$) is a possible solution though, corresponding to a transient chain at ∞.

Suppose ${\varphi }_{\beta }\left(z\right)=1-\lambda {\left(1-z\right)}^{\theta }+o\left(\left({\left(1-z\right)}^{\theta }\right)\right)$ with $\lambda ,\theta \in \left(0,1\right)$, so that ${\mu }_{\beta }:={\varphi }_{\beta }^{\prime }\left(1\right)=\mathbf{E}\left(\beta \right)=\infty .$ Then, as $z\to 1$

$\begin{array}{ccc}\hfill \mathcal{N}\left(z\right)& \sim & {\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)\left(1-z\right)\hfill \\ \hfill \mathcal{D}\left(z\right)& \sim & \alpha \lambda {\left(1-z\right)}^{\theta }\hfill \\ \hfill {\mathrm{\Phi }}_{\infty }\left(z\right)& \sim & {\displaystyle \frac{{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)}{\alpha \lambda }}{\left(1-z\right)}^{1-\theta }\to 0\hfill \end{array}$

The chain asymptotically drifts to ∞ with probability 1.

Suppose ${\mu }_{\beta }:={\varphi }_{\beta }^{\prime }\left(1\right)=\mathbf{E}\left(\beta \right)<\infty$. In that case, the numerator $\mathcal{N}$ and denominator $\mathcal{D}$ both tend to 0 as $z\to 1$ while their ratio ${\mathrm{\Phi }}_{\infty }\left(z\right)$ tends to some constant $C.$ Indeed, as $z\to 1$,

$\begin{array}{ccc}\hfill \mathcal{N}\left(z\right)& \sim & \overline{\alpha }{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)\left(1-z\right)\left(1-{\mu }_{\beta }\left(1-z\right)\right)\hfill \\ \hfill \mathcal{D}\left(z\right)& \sim & \left(1-z\right)\left(\overline{\alpha }-\alpha {\mu }_{\beta }\right)\hfill \\ \hfill {\mathrm{\Phi }}_{\infty }\left(z\right)& \sim & C:=\overline{\alpha }{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)/\left(\overline{\alpha }-\alpha {\mu }_{\beta }\right)={\displaystyle \frac{{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)}{1-{\mu }_{\beta }/{\mu }_{\delta }}}\hfill \end{array}$

With ^′ denoting derivative with respect to z, the L’Hospital rule yields ${\displaystyle \frac{{\mathcal{N}}^{\prime }\left(z\right)}{{\mathcal{D}}^{\prime }\left(z\right)}}\to {\displaystyle \frac{{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)}{1-{\mu }_{\beta }/{\mu }_{\delta }}}$ as $z\to 1$ and does not allow us to fix ${\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)$.

There is, however, a proper p.g.f. solution to (5) if and only if ${\mu }_{\beta }<{\mu }_{\delta }$ and ${\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)=1-{\mu }_{\beta }/{\mu }_{\delta }$. We shall call this regime the subcritical regime. Indeed,

(6)${\mathrm{\Phi }}_{\infty }\left(0\right)=\pi \left(0\right)=b_0{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)=b_0\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right)$

is then the well-behaved probability that the chain is asymptotically in state $\left\{0\right\}$, so that with ${\tau }_{0,0}$, the first return time is to 0 by Kac’s theorem [19]

$\mathbf{E}\left({\tau }_{0,0}\right)=1/\pi \left(0\right)<\infty .$

Plugging this particular value of ${\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)$ into (5), we obtain

${\mathrm{\Phi }}_{\infty }\left(z\right)={\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right){\varphi }_{\beta }\left(z\right){\displaystyle \frac{\overline{\alpha }}{1-\alpha \frac{1-z{\varphi }_{\beta }\left(z\right)}{1-z}}}=:\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right){\displaystyle \frac{\overline{\alpha }{\varphi }_{\beta }\left(z\right)}{1-\alpha {\overline{\varphi }}_{\beta }\left(z\right)}}$

where ${\overline{\varphi }}_{\beta }\left(z\right):={\displaystyle \frac{1-z{\varphi }_{\beta }\left(z\right)}{1-z}}$ is the generating function (g.f.) of the tail probabilities of the $\beta +1$’s, the shifted $\beta$’s by one unit with ${\overline{\varphi }}_{\beta }\left(1\right)=1+{\mu }_{\beta }$.

Theorem 1.

The chain $X_n$ (2) with geometric shrinkage S is ergodic (positive recurrent) if and only if ${\mu }_{\beta }<{\mu }_{\delta }<\infty .$ With $a:=\alpha {\overline{\varphi }}_{\beta }\left(1\right)\in \left(0,1\right)$ ($\overline{a}=1-a$), independently of $X_0$, the p.g.f. ${\mathrm{\Phi }}_{\infty }\left(z\right)$ of $X_{\infty }$ admits the final expression:

(7) ${\mathrm{\Phi }}_{\infty }\left(z\right)={\varphi }_{\beta }\left(z\right){\displaystyle \frac{1-\alpha {\overline{\varphi }}_{\beta }\left(1\right)}{1-\alpha {\overline{\varphi }}_{\beta }\left(z\right)}}={\varphi }_{\beta }\left(z\right){\displaystyle \frac{\overline{a}}{1-aC\left(z\right)}},$

where ${\displaystyle \frac{\overline{a}}{1-aC\left(z\right)}}$ is a compound shifted-geometric $\left(a\right)$ of the r.v.’s with compounding p.g.f. $C\left(z\right)={\overline{\varphi }}_{\beta }\left(z\right)/{\overline{\varphi }}_{\beta }\left(1\right).$

The compounding p.g.f. $C\left(z\right)$ is the delay p.g.f. of $\beta +1$. It has mean $C^{\prime }\left(1\right)=\mathbf{E}\left[\beta \left(\beta +1\right)\right]/\left[2\mathbf{E}\left(\beta +1\right)\right]<\infty$ if and only if $\mathbf{E}\left[{\beta }^2\right]<\infty .$ From (7), $X_{\infty }$ is the sum of two independent components: one is $\beta$, the other one is the r.v. with p.g.f. $\overline{a}/\left(1-aC\left(z\right)\right)$, so $X_{\infty }$ exceeds $\beta$ and if the r.v. $\beta$ is infinitely divisible (compound Poisson), and so is the r.v. $X_{\infty }$. In this positive recurrent regime, we have:

$m:=\mathbf{E}\left(X_{\infty }\right)=\mathbf{E}\left(\beta \right)+{\displaystyle \frac{a}{\overline{a}}}{C}^{\prime }\left(1\right)={\mu }_{\beta }+{\displaystyle \frac{{\mu }_{\delta }\mathbf{E}\left[\beta \left(\beta +1\right)\right]}{2\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right)}}\le \infty$

Inputs $\beta$ with infinite variance yield an invariant probability measure with infinite mean. An example of such an input $\beta$ with infinite variance is the one with p.g.f.

${\varphi }_{\beta }\left(z\right)=1-{\mu }_{\beta }\left(1-z\right)+{\displaystyle \frac{\lambda }{\theta }}{\left(1-z\right)}^{\theta },$

with $\lambda \in \left(0,1\right)$ and $\theta \in \left(1,2\right).$ In this case, ${\overline{\varphi }}_{\beta }\left(z\right)=1+{\mu }_{\beta }z-{\displaystyle \frac{\lambda z}{\theta }}{\left(1-z\right)}^{\theta -1}$ and $C\left(z\right)={\overline{\varphi }}_{\beta }\left(z\right)/{\overline{\varphi }}_{\beta }\left(1\right)$ obeys $C^{\prime }\left(1\right)=\infty .$

No Invariant Measure in the Null-Recurrent Case (Critical Regime)

Null-recurrent or transient random walks on a countable state–space may have or not a stationary measure [20,21]. If it has, it may not be unique. Let $\left[z^x\right]{\mathrm{\Phi }}_{\infty }\left(z\right)$ be the $z^x$-coefficient of ${\mathrm{\Phi }}_{\infty }\left(z\right)$ in its series expansion. If ${\mu }_{\beta }={\mu }_{\delta }$, the critical chain is null-recurrent. Recalling ${\mathrm{\Phi }}_{\infty }\left(z\right)={\displaystyle \frac{\overline{\alpha }{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right){\varphi }_{\beta }\left(z\right)\left(1-z\right)}{1-z-\alpha \left(1-z{\varphi }_{\beta }\left(z\right)\right)}}$ with ${\mathrm{\Phi }}_{\infty }\left(0\right)=\pi \left(0\right)=b_0{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right),$

$\pi \left(x\right):={\mathbf{P}}_{x_0}\left(X_{\infty }=x\right)=\left[z^x\right]{\mathrm{\Phi }}_{\infty }\left(z\right)=\pi \left(0\right){\displaystyle \frac{\overline{\alpha }}{b_0}}\left[z^x\right]{\displaystyle \frac{{\varphi }_{\beta }\left(z\right)\left(1-z\right)}{1-z-\alpha \left(1-z{\varphi }_{\beta }\left(z\right)\right)}},x\ge 1.$

so that

$\pi \left(0\right)=b_0\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right)=0\Rightarrow \pi \left(x\right)=0\mathrm{for}\mathrm{all}x\ge 1.$

The chain (2) has no non trivial ($\ne \mathbf{0}$) invariant positive measure.

2.4. The Generating Functional of the Truncated Geometric Model

With $X_0=x_0\ge 1$ fixed, defining the double generating function

${\mathrm{\Phi }}_{x_0}\left(u,z\right)=\sum _{n\ge 0}{u}^n{\mathrm{\Phi }}_n\left(z\right)=\sum _{n\ge 0}{u}^n{\mathbf{E}}_{x_0}\left(z^{X_n}\right),$

from (4), we obtain

$\left(\overline{\alpha }-z\right){\mathrm{\Phi }}_{x_0}\left(u,z\right)=z^{x_0}\left(\overline{\alpha }-z\right)+u\left[{\varphi }_{\beta }\left(z\right)\left(\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)-\alpha z{\mathrm{\Phi }}_{x_0}\left(u,z\right)\right)\right]$

(8)${\mathrm{\Phi }}_{x_0}\left(u,z\right)={\displaystyle \frac{z^{x_0}\left(\overline{\alpha }-z\right)+\overline{\alpha }u\left(1-z\right){\varphi }_{\beta }\left(z\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)}{\overline{\alpha }-z+\alpha uz{\varphi }_{\beta }\left(z\right)}}.$

We have,

(9)${\mathrm{\Phi }}_{x_0}\left(u,0\right)=G_{x_0,0}\left(u\right)=u{b}_0{\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right),$

${\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)={\mathrm{\Phi }}_{x_0}\left(u,0\right)/\left[u{b}_0\right].$

So far, ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ is unknown since it requires the knowledge of ${\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)$ or of ${\mathrm{\Phi }}_{x_0}\left(u,0\right)$. ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ has a singularity at

(10)$u\left(z\right)=\left(z-\overline{\alpha }\right)/\left[\alpha z{\varphi }_{\beta }\left(z\right)\right].$

In the recurrent regime (${\mu }_{\beta }\le {\mu }_{\delta }$), with $u^{\prime }\left(\overline{\alpha }\right)=1/\left(\alpha \overline{\alpha }{\varphi }_{\beta }\left(\overline{\alpha }\right)\right)>1$, $u\left(z\right)$ is concave and monotone increasing on the interval $\left[\overline{\alpha },1\right],$ with $u^{\prime }\left(1\right)={\mu }_{\delta }-{\mu }_{\beta }\ge 0$ (this can be checked from Proposition 4 stating that its inverse is absolutely monotone as a p.g.f., in particular increasing and convex). The corresponding range of $u\left(z\right)$ is $\left[0,1\right]$. The function $u\left(z\right)$ has thus a well-defined inverse $z\left(u\right)$ which maps $\left[0,1\right]$ to $\left[\overline{\alpha },1\right]$; this inverse is monotone increasing and convex on this interval. We call it the ‘key’ function due to its fundamental interest in the sequel.

When in the recurrent regime, $\left[u^n\right]{\mathrm{\Phi }}_{x_0}\left(u,z\right)\to {\mathrm{\Phi }}_{\infty }\left(z\right)$ as $n\to \infty$, ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ should converge as $z\searrow z\left(u\right)$ for all $u\in \left[0,1\right).$ So both the numerator $\mathcal{N}$ and the denominator $\mathcal{D}$ of ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ must tend to 0 as $z\searrow z\left(u\right)$, meaning (by the L’Hospital rule) that

$\underset{z\searrow z\left(u\right)}{lim}{\displaystyle \frac{\mathcal{N}\left(u,z\right)}{\mathcal{D}\left(u,z\right)}}=\underset{z\searrow z\left(u\right)}{lim}{\displaystyle \frac{{\mathcal{N}}^{\prime }\left(u,z\right)}{{\mathcal{D}}^{\prime }\left(u,z\right)}}.$

Near $z=z\left(u\right),$

$\begin{array}{ccc}\hfill \mathcal{N}\left(u,z\right)& =& z^{x_0}\left(\overline{\alpha }-z\right)+\overline{\alpha }\left(1-z\right)u{\varphi }_{\beta }\left(z\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)=:a\left(z\right)+b\left(z\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)\hfill \\ & \sim & a\left(z\left(u\right)\right)+b\left(z\left(u\right)\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)+\left(z-z\left(u\right)\right)\left[a^{\prime }\left(z\left(u\right)\right)+b^{\prime }\left(z\left(u\right)\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)\right]\hfill \\ \hfill \mathcal{D}\left(u,z\right)& =& :c\left(z\right)-d\left(z\right)u\sim \left(z-z\left(u\right)\right)\left[c^{\prime }\left(z\left(u\right)\right)-d^{\prime }\left(z\left(u\right)\right)u\right],\hfill \end{array}$

imposing $a\left(z\left(u\right)\right)+b\left(z\left(u\right)\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)=0$. Therefore,

${\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }\right)={\displaystyle \frac{z{\left(u\right)}^{x_0}\left(z\left(u\right)-\overline{\alpha }\right)}{\overline{\alpha }u\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}}$

and, from (9),

(11)${\mathrm{\Phi }}_{x_0}\left(u,0\right)=b_0{\displaystyle \frac{z{\left(u\right)}^{x_0}\left(z\left(u\right)-\overline{\alpha }\right)}{\overline{\alpha }\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}}=:G_{x_0,0}\left(u\right)\ge 0,$

is the Green kernel of this model, for $u\in \left[0,1\right];$ with $x_0\ge 1,$${\mathbf{P}}_{x_0}\left(X_0=0\right)=0$ and

$G_{x_0,0}\left(u\right)=\sum _{n\ge 1}{u}^n{\mathbf{P}}_{x_0}\left(X_n=0\right)=\sum _{n\ge 1}{u}^n{P}^n\left(x_0,0\right)={\left(I-uP\right)}^{-1}\left(x_0,0\right),$

is the Green kernel (resolvent) of the chain at the endpoints $\left(x_0,0\right),$ [22]. The matrix element $P^n\left(x_0,0\right)$ is the contact probability at 0 at time n, starting from $x_0\ge 1$. Note from (11) that $G_{x_0,0}\left(1\right)=\infty$ in the recurrent regime, translating that $X_n$ visits state 0 infinitely often.

From (11) and (8), we thus get a closed form expression of ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ when $x_0\ge 1,$ as:

Proposition 2.

In the recurrent regime (${\mu }_{\beta }\le {\mu }_{\delta }$), with $z\left(u\right)$ the inverse of $u\left(z\right)$ defined in (10) and explicit in (17) below, for $u\in \left[0,1\right)$ and $z\ge z\left(u\right),$

(12) ${\mathrm{\Phi }}_{x_0}\left(u,z\right)={\displaystyle \frac{z^{x_0}\left(\overline{\alpha }-z\right)\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)-z{\left(u\right)}^{x_0}\left(\overline{\alpha }-z\left(u\right)\right)\left(1-z\right){\varphi }_{\beta }\left(z\right)}{\left(1-z\left(u\right)\right)\left(\overline{\alpha }-z+\alpha uz{\varphi }_{\beta }\left(z\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}}.$

Remark 3.

If $z=1,$${\mathrm{\Phi }}_{x_0}\left(u,1\right)=1/\left(1-u\right)$ and ${\mathrm{\Phi }}_{x_0}\left(0,z\right)=z^{x_0}$, as required. The generating function ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ in (12) is well defined when $z>z\left(u\right)$ and possibly when $z=z\left(u\right)$ as in the recurrent case.

2.5. First Return Time to 0 (Recurrent Regime)

When $x_0=0$, observing ${\mathbf{P}}_0\left(X_0=0\right)=1,$

(13)$G_{0,0}\left(u\right)=1+{\mathrm{\Phi }}_0\left(u,0\right)=1+{\displaystyle \frac{b_0\left(z\left(u\right)-\overline{\alpha }\right)}{\overline{\alpha }\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}},\mathrm{with}{G}_{0,0}\left(1\right)=\infty (z\left(1\right)=1).$

This function is the Green kernel at the endpoints $\left(0,0\right).$ If $n\ge 1,$ from the recurrence

${\mathbf{P}}_0\left(X_n=0\right)=:P^n\left(0,0\right)=\sum _{m=1}^n\mathbf{P}\left({\tau }_{0,0}=m\right)P^{n-m}\left(0,0\right),$

we see, from taking the generating function of both sides and observing the right hand-side is a convolution, that the p.g.f. ${\varphi }_{0,0}\left(u\right)=\mathbf{E}\left(u^{{\tau }_{0,0}}\right)$ of the first return time to 0, ${\tau }_{0,0}$ and $G_{0,0}\left(u\right)$ are related by the Feller relation (see [23,24], pp. 3–4 for example): $G_{0,0}\left(u\right)=1+$$G_{0,0}\left(u\right){\varphi }_{0,0}\left(u\right)$. Hence, with ${\varphi }_{0,0}\left(0\right)=0$ and ${\varphi }_{0,0}\left(1\right)=1,$

(14)${\varphi }_{0,0}\left(u\right)=\mathbf{E}\left(u^{{\tau }_{0,0}}\right)={\displaystyle \frac{G_{0,0}\left(u\right)-1}{G_{0,0}\left(u\right)}}={\displaystyle \frac{b_0\left(z\left(u\right)-\overline{\alpha }\right)}{b_0\left(z\left(u\right)-\overline{\alpha }\right)+\overline{\alpha }\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}}.$

In particular, observing $z^{\prime }\left(1\right)=1/u^{\prime }\left(1\right)={\displaystyle \frac{1}{{\mu }_{\delta }-{\mu }_{\beta }}}$, we obtain

$\begin{array}{ccc}\hfill \mathbf{E}\left({\tau }_{0,0}\right)& =& \overline{\alpha }{z}^{\prime }\left(1\right)/\left(b_0\alpha \right)={\displaystyle \frac{{\mu }_{\delta }}{b_0\left({\mu }_{\delta }-{\mu }_{\beta }\right)}}\mathrm{if}{\mu }_{\beta }<{\mu }_{\delta }(\mathrm{positive}\mathrm{recurrence}),\hfill \\ & =& \infty \mathrm{if}{\mu }_{\delta }={\mu }_{\beta }(\mathrm{null}\mathrm{recurrence}).\hfill \end{array}$

As required from Kac’s theorem: $\mathbf{E}\left({\tau }_{0,0}\right)=1/\pi \left(0\right)=1/\left[b_0\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right)\right],$ consistent with (6).

2.6. Contact Probability at 0 and First Local Time to Extinction (Recurrent Regime)

With $x_0\ge 1$ (${\varphi }_{x_0,0}\left(0\right)=0$), using (11) and (13),

(15)$\mathbf{E}\left(u^{{\tau }_{x_0,0}}\right)={\varphi }_{x_0,0}\left(u\right)={\displaystyle \frac{G_{x_0,0}\left(u\right)}{G_{0,0}\left(u\right)}}$

$={\displaystyle \frac{b_{0}z{\left(u\right)}^{x_0}\left(z\left(u\right)-\overline{\alpha }\right)}{b_0\left(z\left(u\right)-\overline{\alpha }\right)+\overline{\alpha }\left(1-z\left(u\right)\right){\varphi }_{\beta }\left(z\left(u\right)\right)}}={\varphi }_{0,0}\left(u\right)z{\left(u\right)}^{x_0},$

gives the p.g.f. of the first hitting time of 0, starting from $x_0\ge 1$ (the first local extinction time of the chain). We also obtain

$\begin{array}{ccc}\hfill \mathbf{E}\left({\tau }_{x_0,0}\right)& =& \mathbf{E}\left({\tau }_{0,0}\right)+x_0{z}^{\prime }\left(1\right)={\displaystyle \frac{b_0{x}_0+{\mu }_{\delta }}{b_0\left({\mu }_{\delta }-{\mu }_{\beta }\right)}}\mathrm{if}{\mu }_{\beta }<{\mu }_{\delta }(\mathrm{positive}\mathrm{recurrence}),\hfill \\ & =& \infty \mathrm{if}{\mu }_{\beta }={\mu }_{\delta }(\mathrm{null}\mathrm{recurrence}).\hfill \end{array}$

If $z^{″}\left(1\right)<\infty$ (${\sigma }^2\left(\beta \right)<\infty$ or ${\sigma }^2\left({\tau }_{0,0}\right)<\infty$), the variance of ${\tau }_{x_0,0}$ in the positive-recurrent regime is finite, whereas if ${\sigma }^2\left(\beta \right)=\infty$, ${\sigma }^2\left({\tau }_{x_0,0}\right)=\infty .$

We observe from (10) and (14)

(16)${\varphi }_{0,0}\left(u\right)=\mathbf{E}\left(u^{{\tau }_{0,0}}\right)={\displaystyle \frac{\alpha b_{0}uz\left(u\right)}{\alpha b_{0}uz\left(u\right)+\overline{\alpha }\left(1-z\left(u\right)\right)}},$

so that the laws of ${\tau }_{x_0,0}$ and ${\tau }_{0,0}$ are entirely determined by the key function $z\left(u\right).$

The function $u\left(z\right)$ is explicitly defined in (10) and $z\left(u\right)$ can be obtained from the Lagrange inversion formula as follows: $u\left(z\right)$ has a power series expansion in the scaled variable $\widehat{z}:=\left(z-\overline{\alpha }\right)/\alpha$ and

$u\left(\widehat{z}\right)={\displaystyle \frac{\widehat{z}}{\left(\overline{\alpha }+\alpha \widehat{z}\right){\varphi }_{\beta }\left(\overline{\alpha }+\alpha \widehat{z}\right)}}=:{\displaystyle \frac{\widehat{z}}{{\widehat{\varphi }}_{\beta }\left(\widehat{z}\right)}}$

maps $\left[0,1\right]$ to $\left[0,1\right]$, with $u\left(\widehat{z}=0\right)=0$. By Lagrange inversion formula ([25], p. 159), we can obtain $\widehat{z}\left(u\right)$, satisfying $u\left(\widehat{z}\left(u\right)\right)=u$ as

$\widehat{z}\left(u\right)=\sum _{n\ge 1}{u}^n\left[u^n\right]\widehat{z}\left(u\right)\mathrm{with}\left[u^n\right]\widehat{z}\left(u\right)={\displaystyle \frac{1}{n}}\left[{\widehat{z}}^{n-1}\right]{\widehat{\varphi }}_{\beta }{\left(\widehat{z}\right)}^n.$

Finally, with $\widehat{z}\left(0\right)=0$ and $\widehat{z}\left(1\right)=1$, $z\left(u\right)=\overline{\alpha }+\alpha \widehat{z}\left(u\right)$ and we obtain:

Proposition 3.

It holds that

(17) $z\left(u\right)=\overline{\alpha }+\alpha \sum _{n\ge 1}{\displaystyle \frac{u^n}{n}}\left[{\widehat{z}}^{n-1}\right]{\widehat{\varphi }}_{\beta }{\left(\widehat{z}\right)}^n$

is the series expansion representation of the ’key’ function $z\left(u\right)$. It holds that $\left[u^n\right]z\left(u\right)>0$ translating that $z\left(u\right)$ is a p.g.f.

Remark 4.

One can be more precise. Indeed, $\widehat{z}\left(u\right)$ can be interpreted as the p.g.f. of the total progeny of a Bienaymé–Galton–Watson (BGW) process with branching mechanism p.g.f. ${\widehat{\varphi }}_{\beta }\left(\widehat{z}\right)=\left(\overline{\alpha }+\alpha \widehat{z}\right){\varphi }_{\beta }\left(\overline{\alpha }+\alpha \widehat{z}\right)$ [26] (p. 32); ${\widehat{\varphi }}_{\beta }\left(\widehat{z}\right)$ is thus the p.g.f. of an offspring r.v.

$\nu :=\mathit{ber}\left(\alpha \right)+\sum _{b=0}^{\beta }{\mathit{ber}}_b\left(\alpha \right),$

where ber$\left(\alpha \right)$ is a Bernoulli$\left(\alpha \right)$ r.v. independent of ${\sum }_{b=0}^{\beta }$ber_b$\left(\alpha \right)$ and (ber_b$\left(\alpha \right)$, $b\ge 0$) are i.i.d. Bernoulli$\left(\alpha \right)$, independent of β. Note $\mathbf{E}\left(\nu \right)={\widehat{\varphi }}_{\beta }^{\prime }\left(1\right)<1$ (the subcritical BGW case) if and only if ${\mu }_{\beta }<{\mu }_{\delta }$ in which case ${\widehat{z}}^{\prime }\left(1\right)=1/\left(1-\mathbf{E}\left(\nu \right)\right)<\infty .$

Finally, $z\left(u\right)$ is a zero-inflated p.g.f. version of $\widehat{z}\left(u\right)$. Note also that $z\left(u\right)$ is always a p.g.f., regardless of whether the chain (2) is recurrent or transient. When the chain is recurrent, the inverse $u\left(z\right)$ of $z\left(u\right)$ is increasing and concave on its full definition domain $\left[\overline{\alpha },1\right]$ reaching 1 for the first time at $u=1$, whereas, as sketched below, in the transient regime, $u\left(z\right)$ is increasing and concave only on the sub-interval $\left[\overline{\alpha },z_{*}\right]\subset \left[\overline{\alpha },1\right]$ with $u^{\prime }\left(z_{*}\right)=0$ and $u\left(z_{*}\right)>1.$ On the sub-interval $\left[z_{*},1\right]$, $u\left(z\right)$ is decreasing and concave, with $u^{\prime }\left(1\right)={\mu }_{\delta }-{\mu }_{\beta }<0.$ In that case, $z\left(u\right)$ is the inverse of the left branch expansion of $u\left(z\right)$ near $\overline{\alpha }$.

The function ${\varphi }_{0,0}\left(u\right)$ in (16) is thus a p.g.f., entirely determined by the p.g.f. $z\left(u\right)$. And $z{\left(u\right)}^{x_0}$ is the p.g.f. of the r.v.

(18)${\tau }_{x_0}=\sum _{k=1}^{x_0}{\tau }_k,$

where ${\tau }_k$ is an i.i.d. sequence with ${\tau }_k\stackrel{d}{\sim }\tau$. Note $\mathbf{E}\left(\tau \right)={\displaystyle \frac{\alpha }{\overline{\alpha }}}\mathbf{E}\left({\tau }_{0,0}\right)=\mathbf{E}\left({\tau }_{0,0}\right)/{\mu }_{\delta }$. We have thus completed the proof.

Theorem 2.

In the recurrent regime, the function $z\left(u\right)$ is the p.g.f. of some random variable $\tau \ge 0$. With ${\tau }_{x_0}$ defined in (18), and the first local time to extinction random variable ${\tau }_{x_0,0}$ is decomposable as

(19) ${\tau }_{x_0,0}\stackrel{d}{=}{\tau }_{x_0}+{\tau }_{0,0},$

where ${\tau }_{x_0}\ge 0$ and ${\tau }_{0,0}\ge 1$ are independent. With ${\varphi }_{0,0}\left(u\right)$ given by (16), the p.g.f. of ${\tau }_{x_0,0}$ is given in (15) with $z\left(u\right)$ the inverse (probability-generating) function of $u\left(z\right)=\left(z-\overline{\alpha }\right)/\left[\alpha z{\varphi }_{\beta }\left(z\right)\right]$.

In particular, if ${\sigma }^2\left(\beta \right)<\infty ,$ with ${\sigma }^2\left(\tau \right)={\displaystyle \frac{\alpha }{\overline{\alpha }}}{\sigma }^2\left({\tau }_{0,0}\right)+{\displaystyle \frac{\alpha }{{\overline{\alpha }}^2}}\mathbf{E}{\left({\tau }_{0,0}\right)}^2<\infty ,$

${\sigma }^2\left({\tau }_{x_0,0}\right)=x_0{\sigma }^2\left(\tau \right)+{\sigma }^2\left({\tau }_{0,0}\right)$

Remark 5.

In the recurrent regime, ${\tau }_{x_0,0}$ is only the first local time to extinction (the first hitting time of 0); the chain (2) is being bounced back in the bulk of its definition domain when it hits 0, and there are infinitely many subsequent local times to extinction (separating consecutive excursions). If the chain were forced to have state 0 absorbing, this first local time to extinction would become the ultimate time to extinction.

2.7. The Transient (Supercritical) Regime

In the transient regime (${\mu }_{\beta }>{\mu }_{\delta }$), the denominator of ${\mathrm{\Phi }}_{x_0}\left(u\left(z\right),z\right)$ tends to 0 as $u\to u\left(z\right)$ and $u\left(z\right)$ does not cancel the numerator: $u\left(z\right)$ is a true pole of ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$. When $z\in \left[\overline{\alpha },1\right]$, $u\left(z\right)$ is concave but it has a maximum $u\left(z_{*}\right)$ strictly larger than 1, attained at some $z_{*}$ inside $\left(\overline{\alpha },1\right)$, owing to $u\left(\overline{\alpha }\right)=0,$$u\left(1\right)=1$ and $u^{\prime }\left(1\right)<0$. We anticipate that for some constant $C>0,$

${\mathbf{P}}_{x_0}\left(X_n=0\right)\sim C·u{\left(z_{*}\right)}^{-n}\mathrm{as}n\to \infty ,$

stating that $\left\{X_n\right\}$ only visits 0 a finite number of times before drifting to ∞.

2.7.1. No Invariant Measure in the Transient Regime

Before turning to this question, let us observe the following: suppose ${\mu }_{\beta }>{\mu }_{\delta }$ so that the super-critical geometric chain is transient. If an invariant measure existed, it should satisfy $\pi \left(0\right)=b_0\left(1-{\mu }_{\beta }/{\mu }_{\delta }\right)<0$. The chain has no non-trivial ($\ne \mathbf{0}$) invariant positive measure either. It is not Harris-transient [21].

2.7.2. Large Deviations

Consider $v\left(z\right):=1/u\left(z\right)$ where $u\left(z\right)$, as a pole, cancels the denominator of ${\mathrm{\Phi }}_{x_0}\left(u,z\right)$ without canceling its numerator, so

$v\left(z\right)={\displaystyle \frac{\alpha z{\varphi }_{\beta }\left(z\right)}{z-\overline{\alpha }}}.$

Over the domain $\overline{\alpha }<z\le$ 1, $v\left(z\right)$ is convex with

$v^{\prime }\left(z\right)=\alpha {\displaystyle \frac{{\varphi }_{\beta }^{\prime }\left(z\right)z\left(z-\overline{\alpha }\right)-\overline{\alpha }{\varphi }_{\beta }\left(z\right)}{{\left(z-\overline{\alpha }\right)}^2}}\mathrm{and}{v}^{″}\left(z\right)>0,$

because $z\left(u\right)$ is a p.g.f., its inverse $u\left(z\right)$ is concave. We have

$v\left(1\right)=1\mathrm{and}{v}^{\prime }\left(1\right)={\mu }_{\beta }-{\mu }_{\delta },$

and $v^{\prime }\left(1\right)>0$ if and only if the chain is transient. In this transient regime,

${\mathrm{\Phi }}_n{\left(z\right)}^{1/n}\to v\left(z\right)\mathrm{as}n\to \infty .$

Define the logarithmic generating function

$F\left(\lambda \right):=-logv\left(e^{-\lambda }\right)=logu\left(e^{-\lambda }\right).$

The function $F\left(\lambda \right)$ is concave on its definition domain $\lambda \in \left[0,-log\overline{\alpha }\right)$. Starting from $F\left(0\right)=0$, it is first increasing, attains a maximum, and then decreases to $-\infty$ while crossing zero in between. Therefore, there exists ${\lambda }^{*}\in \left(0,-log\overline{\alpha }\right)$ such that $x_{*}=F^{\prime }\left({\lambda }^{*}\right)=0$. With $x\in \left(F^{\prime }\left(-log\overline{\alpha }\right),F^{\prime }\left(0\right)\right]$, define

(20)$f\left(x\right)=\underset{0\le \lambda <-log\overline{\alpha }}{inf}\left(x\lambda -F\left(\lambda \right)\right)\le 0,$

the Legendre conjugate of $F\left(\lambda \right)$. The variable x is Legendre conjugate to $\lambda$ with $x=F^{\prime }\left(\lambda \right)$ and $\lambda =f^{\prime }\left(x\right)$. Note $F^{\prime }\left(-log\overline{\alpha }\right)=-\infty$ and $F^{\prime }\left(0\right)=v^{\prime }\left(1\right)={\mu }_{\beta }-{\mu }_{\delta }>0.$ On its definition domain, $f\left(x\right)\le 0$ is increasing and concave, starting from $f\left(-\infty \right)=-\infty$ and ending with $f\left(F^{\prime }\left(0\right)\right)=0$ where $f^{\prime }\left(F^{\prime }\left(0\right)\right)=0$. From [27,28], we obtain:

Proposition 4.

For those x in the range $\left[x_{*}=0,F^{\prime }\left(0\right)\right]$ and for any $x_0>0,$

(21) ${\displaystyle \frac{1}{n}}log{\mathbf{P}}_{x_0}\left({\displaystyle \frac{1}{n}}{X}_n\le x\right)\underset{n\to \infty }{error}f\left(x\right).$

In particular, at $x=F^{\prime }\left(0\right)=v^{\prime }\left(1\right)>0$ where $f\left(F^{\prime }\left(0\right)\right)=-F\left(0\right)=0$, we obtain

${\displaystyle \frac{1}{n}}{X}_n\stackrel{a.s.}{\to }r:=v^{\prime }\left(1\right)\mathrm{as}n\to \infty ,$

giving the rate r at which $X_n$ drifts to ∞. To keep $x=F^{\prime }\left(\lambda \right)$ in the non-negative range $\left[x_{*}=0,F^{\prime }\left(0\right)\right]$, the range of $\lambda$ should then equivalently be restricted to $\left[0,{\lambda }^{*}\right]$. We clearly have $f\left(x_{*}\right)=-F\left({\lambda }^{*}\right)<0$, and from (21),

(22)$-{\displaystyle \frac{1}{n}}log{\mathbf{P}}_{x_0}\left({\displaystyle \frac{1}{n}}{X}_n\le 0\right)=-{\displaystyle \frac{1}{n}}log{\mathbf{P}}_{x_0}\left(X_n=0\right)\to F\left({\lambda }^{*}\right)\mathrm{as}n\to \infty .$

This shows the rate at which ${\mathbf{P}}_{x_0}\left(X_n=0\right)$ decays exponentially with n. Equivalently, with $z_{*}=e^{-{\lambda }^{*}}$,

${\mathbf{P}}_{x_0}\left(X_n=0\right)\sim C·u{\left(z_{*}\right)}^{-n},$

as initially guessed.

When ${\mu }_{\beta }>{\mu }_{\delta }$, the number $z_{*}$ inside $\left(\overline{\alpha },1\right)$ exists and it is characterized by $u^{\prime }\left(z_{*}\right)=0$ where $u\left(z\right)=\left(z-\overline{\alpha }\right)/\left[\alpha z{\varphi }_{\beta }\left(z\right)\right],$ hence by

${\displaystyle \frac{{\varphi }_{\beta }^{\prime }\left(z_{*}\right)}{{\varphi }_{\beta }\left(z_{*}\right)}}={\displaystyle \frac{\overline{\alpha }}{z_{*}\left(z_{*}-\overline{\alpha }\right)}},u\left(z_{*}\right)>1.$

When ${\mu }_{\beta }>{\mu }_{\delta }$, the Green series ${\mathrm{\Phi }}_{x_0}\left(1,0\right)=G_{x_0,0}\left(1\right)={\sum }_{n\ge 1}{\mathbf{P}}_{x_0}\left(X_n=0\right)$ is summable for all $x_0\ge 0$ (translating that $X_n$ visits 0 only a finite number of times). In the transient regime, from (15),

$\mathbf{P}\left({\tau }_{x_0,0}<\infty \right)={\varphi }_{x_0,0}\left(1\right)={\displaystyle \frac{G_{x_0,0}\left(1\right)}{G_{0,0}\left(1\right)}}<1.$

The $\beta -$geometric example: Suppose ${\varphi }_{\beta }\left(z\right)=1/\left[1+{\mu }_{\beta }\left(1-z\right)\right].$ Then $z_{*}$ is characterized by the quadratic equation

${\displaystyle \frac{{\mu }_{\beta }}{1+{\mu }_{\beta }\left(1-z_{*}\right)}}={\displaystyle \frac{\overline{\alpha }}{z_{*}\left(z_{*}-\overline{\alpha }\right)}},$

whose positive solution is

$z_{*}=\sqrt{\overline{\alpha }\left(1+1/{\mu }_{\beta }\right)}.$

We have $\overline{\alpha }<z_{*}<1$ if and only if ${\mu }_{\beta }>{\mu }_{\delta }$ (the supercritical regime). In the range $\overline{\alpha }<z<1$, we have

$\begin{array}{ccc}\hfill u^{\prime }\left(z\right)& =& {\displaystyle \frac{\overline{\alpha }\left(1+{\mu }_{\beta }\right)-{\mu }_{\beta }{z}^2}{\alpha z^2}}\hfill \\ \hfill u^{\prime \prime }\left(z\right)& =& -{\displaystyle \frac{2\overline{\alpha }\left(1+{\mu }_{\beta }\right)}{\alpha z^3}}<0,\hfill \end{array}$

showing that $u\left(z\right)$ has the announced properties.

2.7.3. The Scale Function and the Extinction Probability (Transient Case)

In the transient regime, the chain $\left\{X_n\right\}$ started at $x>0$ can drift to ∞ before it first hits 0. There is thus only a probability smaller than 1 that $\left\{X_n\right\}$ gets extinct for the first time.

In the transient regime, let then $P\to P^{\mathrm{abs}}$ with the modification $P^{\mathrm{abs}}\left(0,y\right)={\delta }_{0,y},$ forcing state $\left\{0\right\}$ in P to be absorbing, corresponding to $X_n\to Y_n:=X_{n\wedge {\tau }_{x,0}}$. The matrix $P^{\mathrm{abs}}$ is stochastic but non irreducible, having state $\left\{0\right\}$ as an absorbing class. The harmonic (or scale) sequence $\mathbf{h}$ solves:

(23)$P^{\mathrm{abs}}\mathbf{h}=\mathbf{h},$

where $\mathbf{h}={\left(h\left(0\right),h\left(1\right),...\right)}^{\prime }$ is a column vector. With $y\gg 1$ and $x\in \left\{1,...,y-1\right\},$ let

$\begin{array}{ccc}\hfill {\tau }_{x,y}& =& inf\left(n\ge 0:X_n\ge y\mid X_0=x\right)\mathrm{if}\mathrm{such}\mathrm{an}n\mathrm{exists},\hfill \\ & =& +\infty \mathrm{if}\mathrm{not}.\hfill \end{array}$

By induction, with $h\left(0\right)=1$ and with ${\tau }_x^{\left(y\right)}:={\tau }_{x,0}\wedge {\tau }_{x,y}$

$\forall x\in \left\{1,...,y-1\right\},\forall n\ge 0:h\left(x\right)=\mathbf{E}h\left(Y_{n\wedge {\tau }_{x,y}}\right)=\mathbf{E}h\left(X_{n\wedge {\tau }_x^{\left(y\right)}}\right).$

The harmonic function on $\left\{1,...,y\right\}$, makes $h\left(X_{n\wedge {\tau }_x^{\left(y\right)}}\right)$ a martingale [29]. As $n\to \infty ,$$\forall x\in \left\{1,...,y-1\right\}$:

(24)$h\left(x\right)=\mathbf{E}h\left(X_{{\tau }_x}\right)=h\left(0\right)\mathbf{P}\left({\tau }_{x,0}<{\tau }_{x,y}\right)+\mathbf{E}h\left(X_{{\tau }_{x,y}}\right)\mathbf{P}\left({\tau }_{x,y}<{\tau }_{x,0}\right).$

As $y\to \infty$, both ${\tau }_{x,y}$ and the overshoot $X_{{\tau }_{x,y}}\to \infty$. Assuming there is a solution $h\left(x\right)>0$ such that $h\left(x\right)\to 0$ as $x\to \infty$ yields

(25)$\mathbf{P}\left({\tau }_{x,0}<{\tau }_{x,\infty }\right)=h\left(x\right),$

indeed consistent with the guess $h\left(x\right)>0$ and vanishing at ∞ when $x\to \infty$. This expression also shows that $h\left(x\right)$ must be decreasing with x. Note $\mathbf{P}\left({\tau }_{x,0}<{\tau }_{x,\infty }\right)=\mathbf{P}\left(X_{{\tau }_x}=0\right)$ where ${\tau }_x:={\tau }_{x,0}\wedge {\tau }_{x,\infty }.$ We obtained:

Proposition 5.

In the transient regime, with $\mathbf{h}$ defined in (23) obeying $h\left(0\right)=1$,

$h\left(x\right)=\mathbf{P}\left({\tau }_{x,0}<{\tau }_{x,\infty }\right),$

is the probability of extinction starting from state x.

Remark 6.

The function $\overline{\mathbf{h}}:=\mathbf{1}-\mathbf{h}$ is also clearly a (increasing) harmonic function and so is any convex combination of $\mathbf{h}$ and $\overline{\mathbf{h}}$ with weights summing to 1 [22]. Clearly, $\overline{h}\left(x\right)=\mathbf{P}\left({\tau }_{x,\infty }<{\tau }_{x,0}\right)=\mathbf{P}\left(X_{{\tau }_x}=\infty \right).$

3. Related Random Walks

We introduce and analyze two related random walks on the non-negative integers.

3.1. Life Annuities Policy

Consider now the Markov chain with state-space ${\mathbb{N}}_0$ and with temporal evolution:

(26)$X_{n+1}=X_n\wedge {\delta }_{n+1}+{\beta }_{n+1};X_0>0,\mathrm{else}$

$\mathrm{\Delta }{X}_n=X_{n+1}-X_n=-{\left(X_n-{\delta }_{n+1}\right)}_{+}+{\beta }_{n+1}.$

One of its notable observables is

$Y_{n+1}=-{\left(X_n-{\delta }_{n+1}\right)}_{+},$

the number of individuals that potentially could leave the system.

As before, indeed, $X_n$ may represent the size of some population facing steady demands (emigration) and supplies (immigration). But here, if the demand exceeds $X_n$, there is no update of the population size, whereas if not, the population size is instantaneously switched to the lower size of the demand. As a result, $X_n$ now represents the potential capacity of response of a population (as the amount of individuals it would potentially be able to release) when facing the demands and in the presence of additive supplies. In this sense, (26) is ‘dual’ to (2).

Alternatively, $X_n$ may represent life annuities indexed and ceilinged on the stock price ${\delta }_{n+1}$ of some commodity balanced with an additive automatic penalization itself indexed on the stock price ${\beta }_{n+1}$ of some other commodity, in a stable and stationary environment. As before, competing depletion and growth mechanisms can occur simultaneously.

In (26), the i.i.d. sequence ${\beta }_n$ is again assumed to be independent of ${\delta }_n$ (i.i.d.) and $X_{n-k}$ for all $k\ge 1.$ The Markov chain (26) was recently introduced in [10], but without the interpretations we give here of this model.

Remark 7.

(26) suggests the continuous-time version of this process

$d{X}_t=-{\left(X_{t_{-}}-{\delta }_t\right)}_{+}·dt+d{N}_t$

where ${\delta }_t\stackrel{d}{=}\delta$ for all $t\ge 0$ is a stationary process and $N_t$ a compound-Poisson process with jumps’ amplitudes $\beta \ge 0.$

The one-step stochastic transition matrix P (obeying $P\mathbf{1}=\mathbf{1}$, where $\mathbf{1}$ is a column vector of ones) of the Markov chain $\left\{X_n\right\}$ in (26) is:

(27)$\begin{array}{c}P\left(x,y\right)=\alpha \sum _{z=0}^{x-1}{\overline{\alpha }}^z{b}_{y-z}+{\overline{\alpha }}^x{b}_{y-x},x\ge 1\mathrm{and}y\ge x\hfill \\ P\left(x,y\right)=\alpha \sum _{z=0}^y{\overline{\alpha }}^z{b}_{y-z},x\ge 1\mathrm{and}y<x.\hfill \\ P\left(x,y\right)=\sum _{z=0}^x{d}_z{b}_{y-z},x,y\ge 0.\hfill \end{array}$

In particular $P\left(x,0\right)=d_0{b}_0$ and $P\left(x,x\right)=\alpha {\sum }_{z=0}^{x-1}{\overline{\alpha }}^z{b}_{x-z}+{\overline{\alpha }}^x{b}_0.$

For any given $X_n=x,$$S\left(x\right):=x\wedge \delta \le x$ [ again with: $S\left(X_n\right)\le X_n$ a.s.] has the law now obtained as follows:

$\begin{array}{ccc}\hfill S\left(x\right)& \sim & {\overline{\alpha }}^x{\epsilon }_x+\sum _{y=0}^{x-1}\alpha {\overline{\alpha }}^y{\epsilon }_y\hfill \\ \hfill \mathbf{E}\left(z^{S\left(x\right)}\right)& =& {\overline{\alpha }}^x{z}^x+\alpha \sum _{y=0}^{x-1}{\overline{\alpha }}^y{z}^y={\overline{\alpha }}^x{z}^x+\alpha {\displaystyle \frac{1-{\left(\overline{\alpha }z\right)}^x}{1-\overline{\alpha }z}}\hfill \end{array}$

leading to the local drift and variance terms

$\begin{array}{ccc}\hfill f\left(x\right)& :& =\mathbf{E}\left(X_{n+1}-x\mid X_n=x\right)=\mathbf{E}\left(S\left(x\right)\right)-x+\mathbf{E}\left(\beta \right)=-x+{\displaystyle \frac{\overline{\alpha }}{\alpha }}\left(1-{\overline{\alpha }}^x\right)+\mathbf{E}\left(\beta \right)\hfill \\ \hfill {\sigma }^2\left(x\right)& :& ={\sigma }^2\left(\left(X_{n+1}-x\right)\mid X_n=x\right)={\displaystyle \frac{\overline{\alpha }}{\alpha }}\left(1-{\overline{\alpha }}^x\right)\left[1+{\displaystyle \frac{\overline{\alpha }}{\alpha }}\left(1+{\overline{\alpha }}^x\right)\right]-2{\displaystyle \frac{\overline{\alpha }}{\alpha }}x{\overline{\alpha }}^x+{\sigma }^2\left(\beta \right).\hfill \end{array}$

As x is large, the walker ‘feels’ a negative stabilizing linear drift $f\left(x\right)\sim -x$, whereas the variance ${\sigma }^2\left(x\right)\sim {\displaystyle \frac{\overline{\alpha }}{{\alpha }^2}}+{\sigma }^2\left(\beta \right)={\sigma }^2\left(\delta \right)+{\sigma }^2\left(\beta \right)$ goes to a constant as $x\to \infty .$ Averaging over $X_n$, we obtain

(28)${\mathrm{\Phi }}_{n+1}\left(z\right):=\mathbf{E}\left(z^{X_{n+1}}\right)={\varphi }_{\beta }\left(z\right)\sum _{x\ge 0}\mathbf{P}\left(X_n=x\right)\left({\overline{\alpha }}^x{z}^x+\alpha {\displaystyle \frac{1-{\left(\overline{\alpha }z\right)}^x}{1-\overline{\alpha }z}}\right)$

$\begin{array}{ccc}& =& {\varphi }_{\beta }\left(z\right)\left({\mathrm{\Phi }}_n\left(\overline{\alpha }z\right)+{\displaystyle \frac{\alpha }{1-\overline{\alpha }z}}\left(1-{\mathrm{\Phi }}_n\left(\overline{\alpha }z\right)\right)\right)\hfill \\ & =& {\varphi }_{\beta }\left(z\right){\displaystyle \frac{\alpha +\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_n\left(\overline{\alpha }z\right)}{1-\overline{\alpha }z}}\hfill \end{array}$

Note ${\mathrm{\Phi }}_{n+1}\left(0\right)=b_0\left(\alpha +\overline{\alpha }{\mathrm{\Phi }}_n\left(0\right)\right)$ entails

${\mathrm{\Phi }}_n\left(0\right)=\mathbf{P}\left(X_n=0\right)=b_0\alpha {\displaystyle \frac{1-{\left(b_0\overline{\alpha }\right)}^n}{1-b_0\overline{\alpha }}}+{\left(b_0\overline{\alpha }\right)}^n{\mathrm{\Phi }}_0\left(0\right)$

$\underset{n\to \infty }{\to }\mathbf{P}\left(X_{\infty }=0\right)=a:={\displaystyle \frac{b_0\alpha }{1-b_0\overline{\alpha }}},$

a well-defined limit. The chain (26) being irreducible and aperiodic, it is sufficient for its positive-recurrence to show that ${\mathrm{\Phi }}_n\left(0\right)$ approaches a positive limit as $n\to \infty$. As a result, we obtain:

Proposition 6.

The chain (26) is positive recurrent (ergodic) with ${\mathrm{\Phi }}_{\infty }\left(1\right)=\mathbf{P}\left(X_{\infty }<\infty \right)=1$, whatever the law of β.

The chain (26) is so strongly and rapidly attracted to $\left\{0\right\}$ that there are no heavy-tailed r.v.’s capable to make it switch to a transient regime, a remarkable stability property of this process. This property may be viewed as a consequence of Proposition 4 in [10], stating that, with ${\delta }_m$ i.i.d. geometric $\left(\alpha \right)—$distributed, the iterate of the shrinking operator obeys

${\wedge }_{m=1}^n\left({\delta }_{m+1}\wedge X_m\right)=D_{n+1}{\wedge }_{m=1}^n{X}_m$

where $D_{n+1}$ is an r.v. with geometric distribution having failure probability $1-{\overline{\alpha }}^n$ tending to 1 geometrically fast.

The limiting p.g.f. ${\mathrm{\Phi }}_{\infty }\left(z\right)$ solves

(29)${\mathrm{\Phi }}_{\infty }\left(z\right)={\varphi }_{\beta }\left(z\right){\displaystyle \frac{\alpha +\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }z\right)}{1-\overline{\alpha }z}}$

If ${\mu }_{\beta }<\infty$, then $m:=\mathbf{E}\left(X_{\infty }\right)<\infty$ with $m={\mu }_{\beta }+{\displaystyle \frac{\overline{\alpha }}{\alpha }}\left(1-{\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }\right)\right)$

In one case and for a specific value of $b_0$, the p.g.f. ${\mathrm{\Phi }}_{\infty }\left(z\right)$ is geometric $\left(a\right)$ with the limiting failure probability $a={\displaystyle \frac{b_0\alpha }{1-b_0\overline{\alpha }}}\in \left(0,1\right)$. This occurs when

${\varphi }_{\beta }\left(z\right)=b_0+{\overline{b}}_0{\displaystyle \frac{\alpha z}{1-\overline{\alpha }z}},$

the p.g.f. of a zero-inflated geometric distribution, provided $b_0=1/\left(1+\overline{\alpha }\right)\in \left(0,1\right)$ in which case $a=\alpha$. Indeed,

${\varphi }_{\beta }\left(z\right)={\displaystyle \frac{1}{1+\overline{\alpha }}}{\displaystyle \frac{1-{\overline{\alpha }}^{2}z}{1-\overline{\alpha }z}}$

with mean ${\mu }_{\beta }=\overline{\alpha }/\left[\alpha \left(1+\overline{\alpha }\right)\right]=\overline{\alpha }/\left(1-{\overline{\alpha }}^2\right)$ and ${\mathrm{\Phi }}_{\infty }\left(z\right)={\displaystyle \frac{\alpha }{1-\overline{\alpha }z}}$ solves (29). ${\mathrm{\Phi }}_{\infty }\left(z\right)$ is a one-parameter geometric solution with failure parameter $\alpha$ and mean $\overline{\alpha }/\alpha$. We note that both ${\varphi }_{\beta }\left(z\right)$ and ${\mathrm{\Phi }}_{\infty }\left(z\right)$ share the same algebraic singularity at $z_b=1/\overline{\alpha }$. Except for this exceptional case, the functional Equation (29) has no known explicit solution. However, using singularity analysis, it is possible to extract a large x expression of $\mathbf{P}\left(X_{\infty }=x\right)$ in the case of algebraic-logarithmic singularity of ${\varphi }_{\beta }\left(z\right).$

A short reminder on singularity analysis [30]. Let $\varphi \left(z\right)$ be any analytic function in the indented domain defined by

$D=\left\{z:\left|z\right|\le z_0,\left|\mathrm{Arg}\left(z-z_c\right)\right|>\pi /2-\eta \right\}$

where $z_c$, $z_0>z_c$, and $\eta$ are positive real numbers. Assume that, with $\sigma \left(x\right)=x^a{\left(logx\right)}^b$ where a and b any real numbers (the singular exponents), we have

(30)$\varphi \left(z\right)\sim C_1+C_2\sigma \left({\displaystyle \frac{1}{1-z/z_c}}\right)\mathrm{as}z\to z_c\mathrm{in}D,$

for some real constants $C_1$ and $C_2.$ Then, if $a\notin \left\{0,-1,-2,...\right\}$ the $z^x$-coefficients in the power-series expansion of $\varphi \left(z\right)$ satisfy

(31)$\left[z^x\right]\varphi \left(z\right)\sim C_2{z}_c^{-x}·{\displaystyle \frac{\sigma \left(x\right)}{x}}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}}\mathrm{as}x\to \infty ,$

where $\mathrm{\Gamma }\left(a\right)$ is the Euler function. $\varphi \left(z\right)$ presents an algebraic-logarithmic singularity at $z=z_c.$—if $a\in \left\{0,-1,-2,...\right\}$ and $b\ne 0,$ the singularity $z=z_c$ is purely logarithmic and

(32)$\left[z^x\right]\varphi \left(z\right)\sim C_{2}b{z}_c^{-x}·{\displaystyle \frac{\sigma \left(x\right)}{xlogx}}{\left({\displaystyle \frac{1}{\mathrm{\Gamma }}}\right)}^{\prime }\left(a\right)\mathrm{as}x\to \infty ,$

involving the derivative of the reciprocal of the Euler gamma function at a.

Thus, for algebraic-logarithmic singularities, the asymptote of the $z^x$ coefficients can be read from the singular behavior of the function $\varphi \left(z\right)$ under study.

Coming back to (29), we observe that if the p.g.f. ${\varphi }_{\beta }\left(z\right)$ has a singularity at some $z_b\ge 1$, it is also a singularity for ${\mathrm{\Phi }}_{\infty }\left(z\right)$. Indeed, the term ${\displaystyle \frac{\alpha +\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }z\right)}{1-\overline{\alpha }z}}{\mid }_{z=1/\overline{\alpha }}=1$ so that $1/\overline{\alpha }$ is not a singularity of ${\mathrm{\Phi }}_{\infty }\left(z\right)$ in (29). The singularity analysis of ${\varphi }_{\beta }\left(z\right)$ can essentially be transfered to ${\mathrm{\Phi }}_{\infty }\left(z\right)$.

Example 1.

Suppose $a>0$, $\sigma \left(x\right)=x^a$ and

${\varphi }_{\beta }\left(z\right)\sim C_1+C_2{\left(1-z/z_c\right)}^{-a}\mathrm{as}z\to z_c>1.$

Then, as $z\to z_c,$

$\begin{array}{ccc}\hfill {\mathrm{\Phi }}_{\infty }\left(z\right)& \sim & \left(C_1+C_2{\left(1-z/z_c\right)}^{-a}\right){\displaystyle \frac{\alpha +\overline{\alpha }\left(1-z_c\right){\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }{z}_c\right)}{\alpha +\overline{\alpha }\left(1-z_c\right)}}\sim C_1+C_2^{\prime }{\left(1-z/z_c\right)}^{-a}\hfill \\ \hfill \left[z^x\right]{\mathrm{\Phi }}_{\infty }\left(z\right)& =& \mathbf{P}\left(X_{\infty }=x\right)\sim C_2^{\prime }{z}_c^{-x}·x^{a-1}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(a\right)}},\hfill \end{array}$

and ${\mathrm{\Phi }}_{\infty }\left(z\right)$ behaves similarly as ${\varphi }_{\beta }\left(z\right)$. The singularity of ${\mathrm{\Phi }}_{\infty }\left(z\right)$ is also the one $z_c>1$ of ${\varphi }_{\beta }\left(z\right)$ because ${\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }{z}_c\right)<1$. As a result, $\mathbf{P}\left(X_{\infty }=x\right)$ and $\mathbf{P}\left(\beta =x\right)$ have similar behaviors for large x. If, with $\sigma \left(x\right)=x^{-a}$, ${\varphi }_{\beta }\left(z\right)=1-\lambda {\left(1-z\right)}^a$, $a,\lambda \in \left(0,1\right)$, where $z_c=1$. Then, as $z\to z_c=1,$${\mathrm{\Phi }}_{\infty }\left(z\right)\sim 1-\lambda {\left(1-z\right)}^a$ with $m=\infty .$ This is because ${\displaystyle \frac{\alpha +\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_{\infty }\left(\overline{\alpha }z\right)}{\alpha +\overline{\alpha }\left(1-z\right)}}\to 1$ as $z\to 1$. Here,

$\left[z^x\right]{\mathrm{\Phi }}_{\infty }\left(z\right)=\mathbf{P}\left(X_{\infty }=x\right)\sim -\lambda x^{-\left(a+1\right)}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(-a\right)}},\mathrm{as}x\to \infty ,$

with a power-law tail. Supposing, with $\sigma \left(x\right)={\left(logx\right)}^{-b},$ $b>0,$

${\varphi }_{\beta }\left(z\right)\underset{z\to zc}{\sim }{C}_1+{\displaystyle \frac{C_2}{{\left(-log\left(1-z/z_c\right)\right)}^b}},$

then ${\mathrm{\Phi }}_{\infty }\left(z\right)$ behaves similarly as ${\varphi }_{\beta }\left(z\right)$ and $\left[z^x\right]{\mathrm{\Phi }}_{\infty }\left(z\right)=\mathbf{P}\left(X_{\infty }=x\right)\sim C_2^{\prime }b{z}_c^{-x}·{\displaystyle \frac{1}{x{\left(logx\right)}^{b+1}}}{\left({\displaystyle \frac{1}{\mathrm{\Gamma }}}\right)}^{\prime }\left(0\right)$; when $z_c=1$, both $\beta ,X_{\infty }$ have logarithmic moments of order smaller than b but no moments of any arbitrary positive order. □

With $X_0=x_0\ge 1$, defining the double generating function

${\mathrm{\Phi }}_{x_0}\left(u,z\right)=\sum _{n\ge 0}{u}^n{\mathrm{\Phi }}_n\left(z\right)=\sum _{n\ge 0}{u}^n{\mathbf{E}}_{x_0}\left(z^{X_n}\right),$

from (28), we obtain

Proposition 7.

${\mathrm{\Phi }}_{x_0}\left(u,z\right)=z^{x_0}+{\displaystyle \frac{u{\varphi }_{\beta }\left(z\right)}{1-\overline{\alpha }z}}\left[{\displaystyle \frac{\alpha }{1-u}}+\overline{\alpha }\left(1-z\right){\mathrm{\Phi }}_{x_0}\left(u,\overline{\alpha }z\right)\right].$

Consequently,

${\mathrm{\Phi }}_{x_0}\left(u,0\right)=G_{x_0,0}\left(u\right)={\displaystyle \frac{b_0\alpha u}{\left(1-u\right)\left(1-b_0\overline{\alpha }u\right)}}={\displaystyle \frac{b_0\alpha u}{1-b_0\overline{\alpha }}}\left({\displaystyle \frac{1}{1-u}}-{\displaystyle \frac{b_0\overline{\alpha }}{1-b_0\overline{\alpha }u}}\right)$

is the Green kernel of this model and it is independent of $x_0.$

Indeed, whatever $x_0$, the only way to move to 0 in one step has probability $b_0\alpha$. The contact probability at 0 is given by

$\left[u^n\right]{\mathrm{\Phi }}_{x_0}\left(u,0\right)={\mathbf{P}}_{x_0}\left(X_n=0\right)={\displaystyle \frac{b_0\alpha }{1-b_0\overline{\alpha }}}\left(1-{\left(b_0\overline{\alpha }\right)}^n\right)\underset{n\to \infty }{error}{\displaystyle \frac{b_0\alpha }{1-b_0\overline{\alpha }}},$

so the generic term of a non-summable series is required, translating that state 0 is visited infinitely often. The one-step probability of reaching 0 being independent of the current state gives

Corollary 1.

The first hitting time ${\tau }_{x_0,0}$ of 0 when starting from $x_0\ge 1$, is independent of $x_0$ and geometrically distributed with failure probability $b_0\alpha$.

3.2. Sub-Critical BGW Processes with Immigration: Massive Depletion

For comparison, we revisit these related bi-stochastic processes where a similar competition between birth and death events holds. In such processes, the shrinkage part of the population size is not due to the facing of external demands of emigrants, rather by internal unbalance when on average the branching number per capita is less than 1 (subcriticality). This depletion mechanism is much stronger than the one in (2) and so the conditions on $\beta$ under which (33) is ergodic are much weaker than the ones in Section 2 model.

Bienyamé–Galton–Watson (BGW) processes with random branching number $\nu$ obeying $\mathbf{P}\left(\nu =0\right)>0$ are very unstable, going either extinct or drifting to ∞ [26]. Almost sure extinction of a (sub-)critical BGW process can be avoided while allowing immigration. Let $\left({\beta }_n;n\ge 1\right)$ be an i.i.d. random sequence of r.v.’s taking values in ${\mathbb{N}}_0$ (the immigrants) and let ${\varphi }_{\beta }\left(z\right)=\mathbf{E}\left(z^{\beta }\right)$ be their common p.g.f. With $X_0=0$, consider the Markov branching process with immigration (BGWI) $X_n$, recursively defined by

(33)$X_{n+1}=\sum _{i=1}^{X_n}{\nu }_i+{\beta }_{n+1},n\ge 0.$

Here the ${\nu }_i$’s are the i.i.d. branching numbers of the underlying BGW process. We let ${\varphi }_{\nu }\left(z\right):=\mathbf{E}\left(z^{\nu }\right)$. With $\left[z^{y^{\prime }}\right]{\varphi }_{\nu }\left(z\right)$ the $z^{y^{\prime }}$-coefficient of ${\varphi }_{\beta }\left(z\right)$, $\left\{X_n\right\}$ has one-step transition matrix ($x,y\in N_0$):

$P\left(x,y\right)=\sum _{y^{\prime }=0}^y\left[z^{y^{\prime }}\right]{\varphi }_{\nu }{\left(z\right)}^x{b}_{y-y^{\prime }}.$

Remark 8.

Associated with (33) is the continuous-time version of this Markov process

$d{X}_t=\sum _{i=1}^{X_{t_{-}}}\left({\nu }_i-1\right)·dt+d{N}_t,$

where $N_t$ a compound-Poisson process with jumps’ amplitudes $\beta .$

We assume the underlying BGW is subcritical viz. ${\mu }_{\nu }:={\varphi }_{\nu }^{\prime }\left(1\right)=\mathbf{E}\left(\nu \right)<1$. With $S\left(x\right):={\sum }_{i=1}^x{\nu }_i$ the shrinking part of the dynamics (33), given $X_n=x$, we now have the local drift and variance terms

$\begin{array}{ccc}\hfill f\left(x\right)& :& =\mathbf{E}\left(X_{n+1}-x\mid X_n=x\right)=\mathbf{E}\left(S\left(x\right)\right)-x+\mathbf{E}\left(\beta \right)=-x\left(1-{\mu }_{\nu }\right)+\mathbf{E}\left(\beta \right)\hfill \\ \hfill {\sigma }^2\left(x\right)& :& ={\sigma }^2\left(\left(X_{n+1}-x\right)\mid X_n=x\right)={\sigma }^2\left(S\left(x\right)\right)+{\sigma }^2\left(\beta \right)=x{\sigma }^2\left(\nu \right)+{\sigma }^2\left(\beta \right)\hfill \\ & \sim & x{\sigma }^2\left(\nu \right)\mathrm{as}x\to \infty \mathrm{if}{\sigma }^2\left(\nu \right)<\infty .\hfill \end{array}$

As x is large, the walker ‘feels’ the negative drift $f\left(x\right)\underset{x\to \infty }{\sim }-x\left(1-{\mu }_{\nu }\right).$ Fluctuations are very large for large x, of order x, and so of the diffusive type.

If the underlying BGW is subcritical, the adjunction of immigrants will prevent its extinction in that the BGWI will in general stabilize to a well-defined random limit. The condition is that [31]

${\int }_0^1{\displaystyle \frac{1-{\varphi }_{\beta }\left(z\right)}{{\varphi }_{\nu }\left(z\right)-z}}dz<\infty ,$

which is met if $\mathbf{E}\left(\nu \right)<1$ and $\mathbf{E}log\left(1+\beta \right)<\infty$ (a very weak existence condition of the log-moments of $\beta$ invalidated only would $\beta$ have extremely heavy logarithmic tails). Letting ${\mathrm{\Phi }}_n\left(z\right)=\mathbf{E}\left(z^{X_n}\right),$

$\begin{array}{ccc}\hfill {\mathrm{\Phi }}_{n+1}\left(z\right)& =& {\varphi }_{\beta }\left(z\right){\mathrm{\Phi }}_n\left({\varphi }_{\nu }\left(z\right)\right),{\mathrm{\Phi }}_0\left(z\right)=1\hfill \\ \hfill {\mathrm{\Phi }}_n\left(z\right)& =& \prod _{m=0}^{n-1}{\varphi }_{\beta }\left({\varphi }_{\nu }^{\circ m}\left(z\right)\right)\hfill \end{array}$

where ${\varphi }^{\circ m}\left(z\right)$ is the m-th iterate of ${\varphi }_{\nu }$ and ${\mathrm{\Phi }}_{\infty }\left(z\right)$ obeys the functional equation ${\mathrm{\Phi }}_{\infty }\left(z\right)={\varphi }_{\beta }\left(z\right){\mathrm{\Phi }}_{\infty }\left({\varphi }_{\nu }\left(z\right)\right).$ Hence, under the above condition, ${\mathrm{\Phi }}_{\infty }\left(z\right)={\prod }_{m=0}^{\infty }{\varphi }_{\beta }\left({\varphi }_{\nu }^{\circ m}\left(z\right)\right)$ is the well-defined (convergent) p.g.f. of $X_{\infty }$, translating that $X_{\infty }$ is an infinite sum of independent r.v.’s with p.g.f.’s ${\varphi }_{\beta }\left({\varphi }_{\nu }^{\circ m}\left(z\right)\right)$, $m\ge 0$, with mean $\mathbf{E}\left(X_{\infty }\right)={\varphi }_{\beta }^{\prime }\left(1\right)/\left(1-{\varphi }_{\nu }^{\prime }\left(1\right)\right)\le \infty .$ Note $X_{\infty }\stackrel{d}{=}\beta +E$ where E is an excess r.v. independent of $\beta .$ Equivalently, $X_{\infty }$ is characterized by ($X_{\infty }^{\prime }$ denoting a statistical copy of $X_{\infty }$)

$X_{\infty }\stackrel{d}{=}\sum _{i=1}^{X_{\infty }^{\prime }}{\nu }_i+\beta ,$

with $\beta$ independent of ${\sum }_{i=1}^{X_{\infty }^{\prime }}{\nu }_i$, translating its generalized self-decomposability property induced by the semi-group generated by ${\varphi }_{\nu }$ (see [32,33], Section V.8).

A famous example is the pure-death case: this is when $\nu \stackrel{d}{\sim }$ Bernoulli$\left(c\right),$ equivalently ${\varphi }_{\nu }\left(z\right)=1-c+cz,$ with $c\in \left(0,1\right)$ the survival probability. In that case (alluded to in the Introduction), $S\left(x\right)\stackrel{d}{\sim }$bin$\left(x,c\right)$. This binomial shrinkage effect is appropriate when the individuals of the current population each die or survive in an independent way with some survival probability c. There is thus a resulting drastic depletion of individuals at each step. The BGW process with this branching mechanism is necessarily subcritical. In that case,

${\varphi }_{\nu }^{\circ m}\left(z\right)=1-c^m+c^{m}z$

is itself a Bernoulli with survival probability $c^m.$ Assuming a discrete-stable $\left(\alpha ,\theta \right)$ distribution for $\beta$, $\alpha \in \left(0,1\right]$, $\theta >0$, [with scale parameter $\theta$ and power-law tails of index $\alpha$], it holds that ${\mathrm{\Phi }}_{\infty }\left(z\right)=e^{-\theta /\left(1-c^{\alpha }\right){\left(1-z\right)}^{\alpha }}$, so $X_{\infty }$ is itself $\alpha —$discrete-stable $\left(\alpha ,\theta /\left(1-c^{\alpha }\right)\right)$, but with scale parameter $\theta /\left(1-c^{\alpha }\right).$ We have $\mathbf{E}\left(X_{\infty }\right)=\infty$ if $\alpha \in \left(0,1\right)$ and $\alpha =1$ is the Poisson $\left(\theta \right)$ special case with $\mathbf{E}\left(X_{\infty }\right)=\theta /\left(1-c\right)<\infty$.—The simple birth and death case (binary fission) is when ${\varphi }_{\nu }\left(z\right)={\pi }_0+{\pi }_{1}z+{\pi }_2{z}^2$, ${\pi }_0+{\pi }_1+{\pi }_2=1$ (subcritical when ${\pi }_2<{\pi }_0$). The linear-fractional case is when ${\varphi }_{\nu }\left(z\right)={\pi }_0+{\overline{\pi }}_0{\displaystyle \frac{\pi z}{1-\overline{\pi }z}}$, $\left({\pi }_0,\pi \right)\in \left(0,1\right)$, ${\overline{\pi }}_0=1-{\pi }_0,$$\overline{\pi }=1-\pi$ (subcritical when ${\overline{\pi }}_0/\pi <1$).

In contrast with the truncated geometric model of Section 2, the ergodicity property of BGWI processes can be extended to an underlying critical BGW (with ${\mu }_{\nu }=1$) if [11]

$\mathbf{E}\left(\beta \right)<\infty \mathrm{and}{\int }_0^1{\displaystyle \frac{1-z}{{\varphi }_{\nu }\left(z\right)-z}}dz<\infty ,$

translating that $\nu$ has an infinite variance, viz. ${\varphi }_{\nu }\left(z\right)\underset{z\to 1}{\sim }+{\displaystyle \frac{\lambda }{\alpha +1}}{\left(1-z\right)}^{\alpha +1}+o\left({\left(1-z\right)}^{\alpha +1}\right)$, with both $\lambda ,\alpha \in \left(0,1\right)$. In that case, the limit probability law of $X_{\infty }$ exists and it is heavy-tailed with index $\alpha$ (so in particular with $\mathbf{E}\left(X_{\infty }\right)=\infty$). If the infinite variance property of the underlying BGW process is not satisfied, the BGWI process is transient and drifts to $\infty .$ Still, a non-finite invariant measure is known to exist in this case [34].

In the supercritical regime (${\mu }_{\nu }>1$) and if ${\mu }_{\beta }<\infty ,$ the BGWI process is transient at ∞ and it can be checked that, as $n\to \infty ,$

$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{n}}{X}_n& \to & 0\mathrm{a}.\mathrm{s}.\mathrm{if}1<{\mu }_{\nu }<\infty \hfill \\ \hfill {\displaystyle \frac{1}{n}}{X}_n& \to & \infty \mathrm{a}.\mathrm{s}.\mathrm{if}{\mu }_{\nu }=\infty .\hfill \end{array}$

If $1<{\mu }_{\nu }<\infty$, the sequence ${\mu }_{\nu }^{-n}{X}_n$ is a converging bounded in $L^1$ non-negative submartingale and so

${\mu }_{\nu }^{-n}{X}_n\to Z<\infty \mathrm{a}.\mathrm{s}.$

where the law of Z is characterized by its Laplace-Stieltjes transform in the main Theorem of [35]. Invariant measures are known to exist in the supercritical as well but they are non-unique [34]. The idea is to come down to the non-supercritical case while allowing defective immigration laws.

4. Concluding Remarks

We first analyzed the recurrence/transience conditions of a Markov chain on the non-negative integers where systematic random immigration events promoting growth were simultaneously balanced with random emigration based on a truncated geometric rule, favoring shrinkage. Two related random walks in the same spirit, but with different collapse rules, were introduced. One is a ‘dual’ version of the latter process, shown to be always stable (recurrent), the other is the classical BGW process with immigration where the origin of the pruning is internal, rather due to subcriticality of the population. Their intrinsic statistical properties were shown to be of a completely different nature: the first truncated geometric model turns out to exhibit a large range of transience (the supercritical regime ${\mu }_{\beta }>{\mu }_{\delta }$), the second dual one is never transient, while a subcritical BGW process with immigration fails to be recurrent only when the immigrants’ law has infinite logarithmic moments, a situation which can be considered as exceptional in concrete population model applications.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Acknowledgments

T.H. acknowledges partial support from the labex MME-DII ( Modèles Mathématiques et Économiques de la Dynamique, de l’ Incertitude et des Interactions), ANR11-LBX-0023-01. This work also benefited from the support of the Chair “ Modélisation Mathématique et Biodiversité” of Veolia-Ecole Polytechnique-MNHN-Fondation X. This work was also funded by CY Initiative of Excellence (grant “Investissements d’Avenir”ANR-16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013.

Conflicts of Interest

The author has no conflicts of interest associated with this paper.

Footnotes

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