The Compensation Approach for Walks With Small Steps in the Quarter Plane

Abstract

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane+2with a step set that is a subset of\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]in the interior of+2. We derive an explicit expression for the generating function which turns out to be non-holonomic, and which can be used to obtain exact and asymptotic expressions for the counting numbers.