Vitali Wachtel

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.

Optimal local Hölder index for density states of superprocesses with (1+β)-branching mechanism

For 0 0. If d 1 + β but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Holder index.

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.

Multifractal analysis of superprocesses with stable branching in dimension one

We show that density functions of a (α,1,β)-superprocesses are almost sure multifractal for α>β+1, β∈(0,1) and calculate the corresponding spectrum of singularities.

Sudden Extinction of a Critical Branching Process in a Random Environment

Let

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

T

Critical Galton-Watson processes: The maximum of total progenies within a large window

be the extinction moment of a critical branching process

Universality of local times of killed and reflected random walks

Z=(Z_{n},n\geq 0)

Local asymptotics for the area of random walk excursions

in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process

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

Z