Denis Denisov

Random Walks in Cones

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.

Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

We consider the sums Sn=ξ1+⋯+ξn of independent identically distributed random variables. We do not assume that the ξs have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that M=sup {Sn,n≥1} 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 the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution FI.

Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which P{S� > x} ∼ P{M� > x} ∼ EP{�1 > x} as x → ∞, where Sis a sum �1 + ... + �� of random sizeand Mis a maximum of partial sums M� = maxn�� Sn. Heren, n = 1, 2, . . . , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case whereis independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where E� > 0 and where the tail ofis comparable with or heavier than that of �, and obtain the asymptotics P{S� > x} ∼ EP{�1 > x} + P{� > x/E�} as x → ∞. 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{Sn > x}/P{�1 > x} which substantially improve Kestens bound in the subclass Sof subexpo-

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

For a distribution F*τ of a random sum Sτ=ξ1+⋯+ξτ of i.i.d. random variables with a common distribution F on the half-line [0, ∞), we study the limits of the ratios of tails as x→∞ (here, τ is a counting random variable which does not depend on {ξn}n≥1). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

Exit times for integrated random walks

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. Assuming that the moment of order 2 + δ is finite, we show that the exact asymptotics for this probability are n−1/4. To show these asymptotics we develop a discrete potential theory for integrated random walks.

On Transience Conditions for Markov Chains

holds for every initial X0. Equation (1.1) implies transience of the set {x : L(x) < N} for every N > 0. As far as we know, general conditions for transience have been studied only in the case of countable Markov chains (i.e., chains with a countable state space X ) and under the additional assumption that the values of jumps are bounded. The most general assertion in this case is seemingly the following theorem of [1, p. 31, Theorem 2.2.7].

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 pj 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 in [18]. We then consider a similar type of graph but on the ‘slab’ Z × 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| × |I| random matrix in the Gaussian unitary ensemble (GUE).

An Exact Asymptotics for the Moment of Crossing a Curved Boundary by an Asymptotically Stable Random Walk

Suppose that

Universality of local times of killed and reflected random walks

\{S_n,\ n\geq0\}

Local asymptotics for the area of random walk excursions

is an asymptotically stable random walk. Let