arXiv subjects

Denis Denisov

Publications and source records attributed to Denis Denisov.

43 records · Page 3Linked to original sources

arXiv2011

Martingale approach to subexponential asymptotics for random walks

Consider the random walk $S_n=ξ_1+...+ξ_n$ with independent and identically distributed increments and negative mean $\mathbf Eξ=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_τ>x)$, where $M_τ=\max_{0\le i<τ}S_i$ and $τ=\min\{n\ge 1: S_n\le 0 \}$.

math.PR↗

arXiv2011

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $α<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a conditioned process which lives on a partial compactification of the Weyl chamber.

math.PR↗

arXiv2010

Limit theorems for a random directed slab graph

We consider a stochastic directed graph on the integers whereby a directed edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$ depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied by Foss and Konstantopoulos, Markov Process and Related Fields, 9, 413-468. We then consider a similar type of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When $I$ is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a $|I| \times |I|$ random matrix in the Gaussian unitary ensemble (GUE).

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↗

arXiv2008

On lower limits and equivalences for distribution tails of randomly stopped sums

For a distribution $F^{*τ}$ of a random sum $S_τ=ξ_1+...+ξ_τ$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty)$, we study the limits of the ratios of tails $\bar{F^{*τ}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $τ$ is a counting random variable which does not depend on $\{ξ_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

math.PR↗

arXiv2007

Asymptotics for first-passage times of Lévy processes and random walks

We study the exact asymptotics for the distribution of the first time $τ_x$ a Lévy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(τ_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf P(τ_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.

math.PR↗

arXiv2007

Lower limits for distributions of randomly stopped sums

We study lower limits for the ratio $\frac{\bar{F^{*τ}}(x)}{\bar F(x)}$ of tail distributions where $ F^{*τ}$ is a distribution of a sum of a random size $τ$ of i.i.d. random variables having a common distribution $F$, and a random variable $τ$ does not depend on summands.

math.PR↗