Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Abstract

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

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Kyo-Shin Hwang 1 and Wensheng Wang 2

Recommended by P. G. L. Leach

1, Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea
2, Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 25 February 2012; Accepted 19 June 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let {_Yi ,_Ji } be a sequence of independent and identically distributed random vectors, and write S(n)=_Y1 +_Y2 +...+_Yn and T(n)=_J1 +_J2 +...+_Jn . Let _Nt =max {n...5;0:T(n)...4;t} the renewal process of _Ji . A continuous time random walk (CTRW) is defined by [figure omitted; refer to PDF] In this setting, _Yi represents a particle jump, and _Ji >0 is the waiting time preceding that jump, so that S(n) represents the particle location after n jumps and T(n) is the time of the n th jump. Then _Nt is the number of jumps by time t>0 , and the CTRW X(t) represents the particle location at time t>0 , which is a random walk subordinated to a renewal process.

It should be mentioned that the subordination scheme of CTRW processes is going back to Fogedby [1] and that it was expanded by Baule and Friedrich [2] and Magdziarz et al. [3]. It should also be mentioned that the theory of subordination holds for nonhomogeneous CTRW processes, that were introduced in the following works: Metzler et al. [4, 5] and Barkai et al. [6].

The CTRW is useful in physics for modeling anomalous diffusion. Heavy-tailed particle jumps lead to superdiffusion, where a cloud of particles spreads faster than the classical Brownian motion, and heavy-tailed waiting times lead to subdiffusion. CTRW models and the associated fractional diffusion equations are important in applications to physics, hydrology, and finance; see, for example, Berkowitz et al. [7], Metzler and Klafter [8], Scalas [9], and Meerchaert and Scalas [10] for more information. In applications to hydrology, the heavy tailed particle jumps capture the velocity irregularities caused by a heterogeneous porous media, and the waiting times model particle sticking or trapping. In applications to finance, the particle jumps are price changes or log returns, separated by a random waiting time between trades.

If the jumps _Yi belong to the domain of attraction of a stable law with index α , (0<α<2) , and the waiting times _Ji belong to the domain of attraction of a stable law with index β , (0<β<1) , Becker-Kern et al. [11] and Meerschaert and Scheffler [12] showed that as c[arrow right]∞ , [figure omitted; refer to PDF] a non-Markovian limit with scaling A(E(ct))=d^cβ/α A(E(t)) , where A(t) is a stable Lévy motion and E(t) is the inverse or hitting time process of a stable subordinator. Densities of the CTRW scaling limit A(E(t)) solve a space-time fractional diffusion equation that also involves a fractional time derivative of order β ; see Meerschaert and Scheffler [13], Becker-Kern et al. [11], and Meerschaert and Scheffler [12] for complete details. Becker-Kern et al. [14], Meerschaert and Scheffler [15], and Meerschaert et al. [16] discussed the related limit theorems for CTRWs based on two time scales, triangular arrays and dependent jumps, respectively. The aim of the present paper is to investigate the laws of the iterated logarithm for CTRWs. We establish Chover-type laws of the iterated logarithm for CTRWs with jumps and waiting times in the domains of attraction of stable laws.

Throughout this paper we will use C to denote an unspecified positive and finite constant which may be different in each occurrence and use "i.o." to stand for "infinitely often" and "a.s." to stand for "almost surely" and " u(x)~v(x) " to stand for " lim u(x)/v(x)=1 ". Our main results read as follows.

Theorem 1.1.

Let {_Yi } be a sequence of i.i.d. nonnegative random variables with a common distribution F , and let {_Ji } , independent of {_Yi } , be a sequence of i.i.d. nonnegative random variables with a common distribution G . Assume that 1-F(x)~^x-α L(x) , 0<α<2 , where L is a slowly varying function, and that G is absolutely continuous and 1-G(x)~C^x-β , 0<β<1 . Let {B(n)} be a sequence such that nL(B(n))/B(n^)α [arrow right]C as n[arrow right]∞ . Then one has [figure omitted; refer to PDF]

The following is an immediate consequence of Theorem 1.1.

Corollary 1.2.

If the tail distribution of _Yi satisfies P(_Y1 >x)~C^x-α in Theorem 1.1, then one has [figure omitted; refer to PDF]

In the course of our arguments we often make statements that are valid only for sufficiently large values of some index. When there is no danger of confusion, we omit explicit mention of this proviso.

2. Chung Type LIL for Stable Summands

In this section we consider a Chung-type law of the iterated logarithm for sums of random variables in the domain of attraction of a stable law, which will take a key role to show Theorem 1.1. When _Ji has a symmetric stable distribution function G characterized by [figure omitted; refer to PDF] 0<β<2 . Chover [17] established that [figure omitted; refer to PDF] We call (2.2) as Chover's law of the iterated logarithm. Since then, several papers have been devoted to develop Chover's LIL; see, for example, Hedye [18-20], Pakshirajan and Vasudeva [21], Vasudeva [22], Qi and Cheng [23], Scheffler [24], Chen [25], and Peng and Qi [26] for reference. For some reason the obvious corresponding statement for the "lim inf" result does not seem to have been recorded, and it is the purpose of this section to do so and may be of independent interest.

Theorem 2.1.

Let {_Ji } be a sequence of i.i.d. nonnegative random variables with a common distribution G(x) , and let V(x)=inf {y>0:1-G(y)...4;1/x} . Assume that G is absolutely continuous and 1-G(x)~^x-β l(x) , 0<β<1 , where l is a slowly varying function. Then one has [figure omitted; refer to PDF]

In order to prove Theorem 2.1, we need some lemmas.

Lemma 2.2.

Let h(x) be a slowly varying function. Then, if _yn [arrow right]∞ , _zn [arrow right]∞ , one has for any given τ>0 , [figure omitted; refer to PDF]

Proof.

See Seneta [27].

Lemma 2.3.

Let {_Ji } be a sequence of i.i.d. nonnegative random variables with a common distribution G and let M(n)=max {_J1 ,_J2 ,...,_Jn } . Assume that G is absolutely continuous and 1-G(x)~^x-β l(x) , 0<β<1 , where l is a slowly varying function. Then one has for some given small t>0 [figure omitted; refer to PDF]

Proof.

We will follow the argument of Lemma 2.1 in Darling [28]. Without loss of generality we can assume _J1 =max {_J1 ,_J2 ,...,_Jn }=M(n) since each _Ji has a probability of 1/n of being the largest term, and P(_Ji =_Jj )=0 for i...0;j since G(x) is presumed continuous.

For notational simplicity we will use the tail distribution G¯(x)=1-G(x)=P(_J1 >x) and denote by g(x) the corresponding density, so that G¯(x)=^∫x∞ g(z)dz . Then, the joint density of _J1 ,_J2 ,...,_Jn , given _J1 =M(n) , is [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Let us put [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] It follows from Doeblin's theorem that if λ>0 , [figure omitted; refer to PDF] for y...5;_y0 with some large _y0 >0 . Then, for y...4;_y0 , we can choose t>0 small enough such that t<-log G(_y0 ) since G has regularly varying tail distribution, so that [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] Consider the case y...5;_y0 . By a slight transformation we find that [figure omitted; refer to PDF] Putting [figure omitted; refer to PDF] we have η<1 since 0<β<1 and t is small. Thus [figure omitted; refer to PDF] By (2.9) and making the change of variable nG¯(y)=v to give [figure omitted; refer to PDF] which yields the desired result.

The following large deviation result for stable summands is due to Heyde [19].

Lemma 2.4.

Let {_ξi } be a sequence of i.i.d. nonnegative random variables with a common tail distribution satisfying P(_ξ1 >x)~^x-r h(x) , 0<r<2 , where h is a slowly varying function. Let {_λn } be a sequence such that nh(_λn )/^λnr [arrow right]C as n[arrow right]∞ , and let {_xn } be a sequence with _xn [arrow right]∞ as n[arrow right]∞ . Then [figure omitted; refer to PDF]

Now we can show Theorem 2.1.

Proof of Theorem 2.1.

In order to show (2.3), it is enough to show that for all [varepsilon]>0 [figure omitted; refer to PDF]

We first show (2.18). Let _nk =[^θk ] , 1<θ<2 . Put again G¯(x)=1-G(x)=P(_J1 >x) . Let ^G¯* be the inverse of G¯ . Obverse that ^G¯* (y)~^y-1/β H(1/y) , 0<y...4;1 , where H is a slowly varying function and V(n)=^G¯* (1/n)~^n1/β H(n) , so that [figure omitted; refer to PDF] by Lemma 2.2. Let U,_U1 ,_U2 ,...,_Un be i.i.d. random variables with the distribution of U Uniform over (0,1) , and let ^M* (n)=max {_U1 ,_U2 ,...,_Un } . Then, from the fact that G(_Jn ) is a Uniform (0,1) random variable, we note that ^M* (n)=dG(M(n)) , n...5;1 . From (2.21), _Ji nonnegative, and G¯ and ^G¯* nonincreasing, it follows that [figure omitted; refer to PDF] Hence, the sum of the left hand side of the previously mentioned probability is finite; by the Borel-Cantelli lemma, we get [figure omitted; refer to PDF] Thus, by (2.20) we have [figure omitted; refer to PDF] Therefore, by the arbitrariness of θ>1 , (2.18) holds.

We now show (2.19). Let _nk =[^ek1+δ ] , δ>0 . For notational simplicity, we introduce the following notations: [figure omitted; refer to PDF] By Lemma 2.3, we have [figure omitted; refer to PDF] Thus, we get ∑...P(_Ok )<∞ .

Observe again that ^G¯* (y)~^y-1/β H(1/y) and V(n)~^n1/β H(n) , so that [figure omitted; refer to PDF] by Lemma 2.2. Thus, we note [figure omitted; refer to PDF] which yields easily ^∑ P(_Fk )=∞ . Hence, since P(_Ek )...5;P(_Fk )-P(_Ok ) , we get ∑...P(_Ek )=∞ . Since _Ek are independent, by the Borel-Cantelli lemma, we get [figure omitted; refer to PDF]

By applying Lemma 2.4 and (2.27) and some simple calculation, we have easily that ∑...P(_E~k )<∞ , so that [figure omitted; refer to PDF] which, together with (2.30), implies [figure omitted; refer to PDF] This yields (2.19). The proof of Theorem 2.1 is now completed.

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

We have to show that for all [varepsilon]>0 [figure omitted; refer to PDF]

We first show (3.1). Let _tk =^θk , 1<θ<2 . For notational simplicity, we introduce the following notations: [figure omitted; refer to PDF]

By (2.18), we have [figure omitted; refer to PDF]

Put F¯(x)=1-F(x)=P(_Y1 >x) . Let ^F¯* be the inverse of F¯ . Recall that ^F¯* (y)~^y-1/α H~(1/y) , 0<y...4;1 , where H~ is a slowly varying function, so that B(n)=^F¯* (C/n)~C^n1/α H~(n) and [figure omitted; refer to PDF]

Note that [figure omitted; refer to PDF] Thus, by noting U increasing, [figure omitted; refer to PDF] Hence, by Lemma 2.2, [figure omitted; refer to PDF] Thus, by (3.8) and Lemma 2.4, we have [figure omitted; refer to PDF] Therefore, ∑...P(_Q~k )<∞ . By the Borel-Cantelli lemma, we get P(_Q~k i.o.)=0 .

Observe that [figure omitted; refer to PDF] where ^Ec stands for the complement of E . Thus, letting n[arrow right]∞ , we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Thus, by (3.5), we have [figure omitted; refer to PDF] This yields (3.1) immediately by letting θ[arrow down]1 .

We now show (3.2). Let _tk =^ek1+δ , δ>0 . To show (3.2), it is enough to prove [figure omitted; refer to PDF]

Put [figure omitted; refer to PDF]

By (2.19), we have [figure omitted; refer to PDF]

Note that [figure omitted; refer to PDF] Thus, by noting _U1 increasing, [figure omitted; refer to PDF] Hence, by Lemma 2.2, [figure omitted; refer to PDF] Similarly, by noting _tk /_tk-1 [arrow right]∞ , one can have [figure omitted; refer to PDF] Thus, by Lemma 2.4, we have [figure omitted; refer to PDF] Therefore, ∑...P(_Wk )=∞ . Since the events {_Wk } are independent, by the Borel-Cantelli lemma, we get P(_Wk i.o.)=1 .

Now, observe that [figure omitted; refer to PDF] Therefore, by letting m[arrow right]∞ , we get [figure omitted; refer to PDF] which implies (3.14). The proof of Theorem 1.1 is now completed.

Remark 3.1.

By the proof Theorem 1.1, (1.3) can be modified as follows: [figure omitted; refer to PDF] That is to say that the form of (1.3) is no rare and the variables (B(^tβ )^)-1 X(t) must be cut down additionally by the factors (log t^)-1/α to achieve a finite lim sup.

Acknowledgments

The authors wish to express their deep gratitude to a referee for his/her valuable comments on an earlier version which improve the quality of this paper. K. S. Hwang is supported by the Korea Research Foundation Grant Funded by Korea Government (MOEHRD) (KRF-2006-353-C00004), and W. Wang is supported by NSFC Grant 11071076.

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