arXiv subjects

D. Korshunov

Publications and source records attributed to D. Korshunov.

3 recordsLinked to original sources

arXiv2019

Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$

For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $\mathbb Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n\mathbb I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erdős), which corresponds to a function $f(i)=\mathbb I\{i\ge 1\}$; (b) $α$-fold self-intersections of the random walk (studied by Becker and König), which corresponds to $f(i)=i^α$; (c) sites visited by the random walk exactly $j$ times (considered by Erdős and Taylor and by Pitt), where $f(i)=\mathbb I\{i=j\}$.

math.PR↗

arXiv2013

Asymptotics for sums of random variables with local subexponential behaviour

We study distributions $F$ on $[0,\infty)$ such that for some $T\le\infty$, $F^{*2}(x,x+T]\sim 2 F(x,x+T]$. The case $T=\infty$ corresponds to $F$ being subexponential, and our analysis shows that the properties for $T<\infty$ are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman-Harris branching processes.

math.PR↗

arXiv2013

Tail asymptotics for the supremum of a random walk when the mean is not finite

We consider the sums $S_n=ξ_1+\cdots+ξ_n$ of independent identically distributed random variables. We do not assume that the $ξ$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${\bf P}\{M>x\}$ as $x\to\infty$, provided that $M=\sup\{S_n,\ n\ge1\}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.

math.PR↗