# Transient phenomena for random walks and their applications

*by Professor Alexander A Borovkov*

*Institution:*Sobolev Institute of Mathematics, Novosibirsk

*Date: Wed 2nd March 2005*

*Time: 11:00 AM*

*Location: Russell Love Theatre, Richard Berry Building, The University of Melbourne*

*Abstract*: Let Z be the global maximum in a random walk {Zn} with independent jumps Xi. It is well known that if the X\'s are identically distributed, then Z is finite almost surely when A=-EX>0, and is infinite almost surely when A=0.

Often in applications one has small values of A (heavy traffic problems of

queueing theory, ruin probabilities for low-profit insurance companies etc). Then an important question is how large Z can be, and what the limiting distribution of the properly normalized Z is as A-->0. The answer to this question in the case when the X\'s are identically distributed and have a finite variance s^2 has been well known since the early 1960\'s (Prokhorov, Kingman):

P (AZ > t) --> exp{-2t/s^2} as A --> 0.

The talk is devoted to the study of the limit distribution of max{Z1,...,Zn} in the important for applications case when:

* the X\'s are non-identially distributed, with finite means Ai=EXi,

* EXi^2 is infinite,

* A = (A1 + ... + An)/n tends to zero.

A function d(A) is found such that for n=Td(A)/A (T is independent of A

and can be infinite) there exists a limiting distribution for max{Z1,...,Zn}/d(A). When T is infinite, the limit distribution is found in explicit form for some special cases.

In conclusion, we discuss some applications of the new asymptotic results to the theory of integral equations of Wiener-Hopf type.

*For More Information:* Paul Norbury, tel: 8344-5534, email p.norbury@ms.unimelb.edu.au