On a walk in a strip with inhibitory boundaries

In the paper we have obtained ergodic theorems for walks generated by sums of stationarily connected random variables and two inhibitory boundaries. We have found representations for the steady-state distributions, permitting us to obtain in the case of independent summands a whole series of useful formulas. We discuss the connection of the results established with problems from queueing theory.     

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Let ξ₁,ξ₂,... be independent random variables with distributions F₁F₂,... in a triangular array scheme (F _i may depend on some parameter). Assume that Eξ _i = 0, Eξ _i ² < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S _n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S _k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max_k≤n(S _k ? g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.     

On the Gibbs Properties of Bernoulli Convolutions Related to β-Numeration in Multinacci Bases

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the -numeration. A matrix decomposition of these measures is obtained in the case when is a PV number. We also determine their Gibbs properties for being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.