Two-sided problem for the random walk with bounded maximal increment

Abstract

Abstract We consider a random walk with zero drift and finite positive variance σ 2. For positive numbers y, z we find the limit as n → ∞ $n\rightarrow\infty$ of the probability that the first exit of the walk from interval - z σ n , y σ n $\left(-z\sigma\sqrt{n}, y\sigma\sqrt{n}\right)$ occurs through its left end, while the maximum increment of the walk until the exit is smaller than x σ n $x\sigma\sqrt{n}$ , where x is a positive number. The limit theorem is established for the moment of the first exit of the walk from the indicated interval under the condition that this exit occurs through its left end and the value of the maximum walk increment is bounded.