Abstract/Details

## SOME PROBLEMS IN PROBABILITY AND ANALYSIS

CHAUDHURI, RANJAN.
University of South Florida ProQuest Dissertations Publishing,  1982. 8229625.

### Abstract (summary)

Let (P(,n)) be an (s x s) convergent non-homogeneous stochastic chain. Then we can associate with (P(,n)) a partition {T,C(,1),C(,2),..,C(,p)} of the set {1,2,..,s} called the basis of the chain. If P(,n)'s are bistochastic, then T = (phi) and it is known that

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

is finite whenever i (epsilon) C(,m) and j (epsilon) C(,n) (m (NOT=) n). Also, in the bistochastic case, each of the smaller chains (P(,n)(C(,i))) defined by

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

and q,j (epsilon) C(,i) is strongly ergodic. This result is also true if (P(,n)) is a purely stochastic chain that is convergent with a basis in which T = (phi). In this dissertation, the above problem is considered for arbitrary, non-homogeneous, convergent, stochastic chains for which the 'T' set in the basis need not be empty. Some conditions sufficient for the above to hold in the case of general chains are given. Also, some conditions necessary for a non-homogeneous stochastic chain (P(,n)) to converge with the basis {T,C(,1),..,C(,p)} are explored in depth.

The second part of this dissertation is concerned with the structure of idempotent boolean matrices. A characterization theorem on the structure of such matrices is given. Some applications of boolean matrices in the context of non-homogeneous Markov chains are also explored briefly in the first part of this thesis.

The final part of this dissertation is concerned with an open problem on probabilistic context-free grammars. It has been proved that the average derivation length and the average word length of the words in a probabilistic context-free language are respectively equal to the mean derivation length and the mean word length of the words in a sample from the language whenever the production probabilities are estimated from the sample.

### Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
SOME PROBLEMS IN PROBABILITY AND ANALYSIS
Author
CHAUDHURI, RANJAN
Number of pages
90
Degree date
1982
School code
0206
Source
DAI-B 43/07, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-204-21909-0
University/institution
University of South Florida
University location
United States -- Florida
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
8229625
ProQuest document ID
303266970