On the number of zero increments of random walks with a barrier

Abstract

Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as $n \to \infty$) behaviour of $V_n$ the number of zero increments before the absorption in a random walk with the barrier $n$. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for $V_n$ under a regular variation assumption.

Keywords

• absorption time
• recursion with random indices
• random walk
• undershoot
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-ds] mathematics [math]/dynamical systems [math.ds]
• [math.math-co] mathematics [math]/combinatorics [math.co]