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Dmitry Korshunov

Publications and source records attributed to Dmitry Korshunov.

At least 19 recordsLinked to original sources
arXiv2026

On extremes of a random walk with positive drift over an intermediate regularly varying time interval

We consider a random walk $\{S_n\}$ with a finite positive drift that is stopped at a random time $τ$ having an intermediate regularly varying distribution. We assume that the jump distribution is lighter-tailed than the distribution of $τ$. Under these conditions, we show that the tails of the distributions of $S_τ$ and $M_τ = \max_{k\le τ} S_k$ are asymptotically equivalent and are determined by the tail of $τ$, while the random walk $\{S_n\}$ contributes only through the law of large numbers.

math.PR↗
arXiv2024

Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a Lévy process, both with negative drift, over random time horizon $τ$ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by $τ$ independent of the processes. We link our results with random walk theory.

math.PR↗
arXiv2023

Probabilistic approach to risk processes with level-dependent premium rate

We study risk processes with level dependent premium rate. Assuming that the premium rate converges, as the risk reserve increases, to the critical value in the net-profit condition, we obtain upper and lower bounds for the ruin probability. In contrast to existing in the literature results, our approach is purely probabilistic and based on the analysis of Markov chains with asymptotically zero drift.

math.PR↗
arXiv2023

At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift

The main goal of this text is comprehensive study of time homogeneous Markov chains on the real line whose drift tends to zero at infinity, we call such processes Markov chains with asymptotically zero drift. Traditionally this topic is referred to as Lamperti's problem. Time homogeneous Markov chains with asymptotically zero drift may be viewed as a subclass of perturbed in space random walks. The latter are of basic importance in the study of various applied stochastic models, among them branching and risk processes, queueing systems etc. Random walks generated by sums of independent identically distributed random variables are well studied, see e.g. classical textbooks by W. Feller (1971), V.V. Petrov (1975), or F. Spitzer (1964); for the recent development of the theory of random walks we refer to A.A. Borovkov and K.A. Borovkov (2008). There are many monographs devoted to various applications where random walks play a crucial role, let us just mention books on ruin and queueing processes by S. Asmussen (2000, 2003); on insurance and finance by P. Embrechts, C. Kluppelberg, and T. Mikosch (1997), and T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels (1998); and on stochastic difference equations by D. Buraczewski, E. Damek and T. Mikosch (2016). In the same applied stochastic models, if one allows the process considered to be dependent on the current state of the process, we often get a Markov chain which has asymptotically zero drift, we demonstrate that in the last chapter, where we particularly discuss branching and risk processes, stochastic difference equations and ALOHA network.

math.PR↗
arXiv2020

Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution $F$ of $ξ_n := \log ((1-A_n)/A_n)$ is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n-th generation which becomes even heavier with increase of n. More precisely, we prove that, for any n, the distribution tail $\mathbb{P}(Z_n > m)$ of the $n$-th population size $Z_n$ is asymptotically equivalent to $n\overline{F}(\log m)$ as $m$ grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved this result in the case $n=1$ for regularly varying environment $F$ with parameter $α>1$. Further, for a subcritical branching process with subexponentially distributed $ξ_n$, we provide the asymptotics for the distribution tail $\mathbb{P}(Z_n>m)$ which are valid uniformly for all $n$, and also for the stationary tail distribution. Then we establish the "principle of a single atypical environment" which says that the main cause for the number of particles to be large is a presence of a single very small environmental parameter $A_k$.

math.PR↗
arXiv2020

Slowly varying asymptotics for signed stochastic difference equations

For a stochastic difference equation $D_n=A_nD_{n-1}+B_n$ which stabilises upon time we study tail distribution asymptotics of $D_n$ under the assumption that the distribution of $\log(1+|A_1|+|B_1|)$ is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the present paper is three-fold. Firstly, we identify the asymptotic behaviour not only of the stationary tail distribution but also of $D_n$. Secondly, we solve the problem in the general setting when $A$ takes both positive and negative values. Thirdly, we get rid of auxiliary conditions like finiteness of higher moments used in the literature before.

math.PR↗
arXiv2020

Renewal Theory for Transient Markov Chains with Asymptotically Zero Drift

We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain $X_n$ in $\mathbf R$, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by $X_n$ in the interval $(x,x+1]$ is roughly speaking the reciprocal of the drift and tends to infinity as $x$ grows. For the first time we present a general approach relying in a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as $1/x$ or much slower than that, say as $1/x^α$ for some $α\in(0,1)$. The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case $X_n^2/n$ converges weakly to a $Γ$-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for $X_n^{1+α}/n$ and further normal approximation is available.

math.PR↗
arXiv2017

Two-dimensional ruin probability for subexponential claim size

We analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity and generic claim size is subexponential.

math.PR↗
arXiv2016

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cramér-Lundberg risk process.

math.PR↗
arXiv2016

A look at perpetuities via asymptotically homogeneous in space Markov chains

It is shown how a natural representation of perpetuities as asymptotically homogeneous in space Markov chains allows to prove various asymptotic tail results for stable perpetuities and limit theorems for unstable ones. Some of these results are new while others essentially improve moment conditions known in the literature. Both subexponential and Cramér's cases are considered.

math.PR↗
arXiv2015

Asymptotic Expansion of Gaussian Chaos via Probabilistic Approach

For a centered $d$-dimensional Gaussian random vector $ξ=(ξ_1,\ldots,ξ_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(ξ)$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(ξ)$ and its density at infinity and then discuss possible extensions for some general $ξ$ with polar representation.

math.PR↗
arXiv2013

Harmonic functions and stationary distributions for asymptotically homogeneous transition kernels on $Z^+$

We suggest a method for constructing positive harmonic functions for a wide class of transition kernels on $Z^+$. We also find natural conditions under which these functions have positive finite limits at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on $Z^+$ with asymptotically negative drift. More precisely, assuming that Markov chain satisfy Cramér's condition, we study the tail asymptotics of the stationary distribution. In particular, we clarify the influence of the rate of convergence of jumps of the chain towards the limiting distribution.

math.PR↗
arXiv2013

Heavy tails in multi-server queues

In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time $W$ in the $GI/GI/2$ FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability ${\bf P}\{W>x\}$. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load.

math.PR↗
arXiv2013

Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, $\E e^{λZ_1}=\infty$ for every $λ>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur.

math.PR↗
arXiv2012

On Large Delays in Multi-Server Queues with Heavy Tails

We present upper and lower bounds for the tail distribution of the stationary waiting time $D$ in the stable $GI/GI/s$ FCFS queue. These bounds depend on the value of the traffic load $ρ$ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a `principle of $s-k$ big jumps' in this case (here $k$ is the integer part of $ρ$), which gives the most probable way for the stationary waiting time to be large. Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.

math.PR↗
arXiv2012

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.

math.PR↗
arXiv2009

Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which $P(S_τ>x)\sim P(M_τ>x)\sim EτP(ξ_1>x)$ as $x\to\infty$, where $S_τ$ is a sum $ξ_1+...+ξ_τ$ of random size $τ$ and $M_τ$ is a maximum of partial sums $M_τ=\max_{n\leτ}S_n$. Here $ξ_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $τ$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $Eξ>0$ and where the tail of $τ$ is comparable with or heavier than that of $ξ$, and obtain the asymptotics $P(S_τ>x) \sim EτP(ξ_1>x)+P(τ>x/Eξ)$ as $x\to\infty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(ξ_1>x)$ which substantially improve Kesten's bound in the subclass ${\mathcal S}^*$ of subexponential distributions.

math.PR↗