We study the asymptotic behaviour 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 walkâ€¦ (More)

Let S 0 = 0, {Sn, n â‰¥ 1} be a random walk generated by a sequence of i.i.d. random variables X 1 , X 2 , ... and let Ï„ âˆ’ = min{n â‰¥ 1 : Sn â‰¤ 0} and Ï„ + = min{n â‰¥ 1 : Sn > 0}. Assuming that theâ€¦ (More)

There is a well-known sequence of constants c n describing the growth of supercritical Galton-Watson processes Z n. With " lower deviation probabilities " we refer to P(Z n = k n) with k n = o(c n)â€¦ (More)

For families of random walks {S k } with ES (a) k = âˆ’ka < 0 we consider their maxima M(a) = supkâ‰¥0 S (a) k . We investigate the asymptotic behaviour of M(a) as a â†’ 0 for asymptotically stable randomâ€¦ (More)

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 thatâ€¦ (More)

In a recent paper of Eichelsbacher and KÃ¶nig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct aâ€¦ (More)

In this paper we prove a conditional limit theorem for a critical Galtonâ€“Watson branching process {Zn; n â‰¥ 0} with offspring generating function s + (1 âˆ’ s)L((1 âˆ’ s)âˆ’1), where L(x) is slowly varying.â€¦ (More)

Consider a random walk Sn = âˆ‘ni=0 Xi with negative drift. This paper deals with upper bounds for the maximum M = maxnâ‰¥1 Sn of this random walk in different settings of power moment existences. As isâ€¦ (More)

This paper continues our previous work [4] 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 thatâ€¦ (More)

Let T be the extinction moment of a critical branching process Z = (Zn, n â‰¥ 0) in a random environment specified by iid probability generating functions. We study the asymptotic behavior of theâ€¦ (More)