Weak convergence of the number of zero increments in the random walk with barrier

TL;DR

This paper proves that the properly normalized count of zero increments in a specific type of random walk with a barrier converges weakly to a normal distribution, refining earlier weak law results.

Contribution

It establishes the weak convergence of zero increment counts in random walks with barriers under power-like tail assumptions, extending previous law of large numbers results.

Findings

Weak convergence to normal law for zero increments

Refinement of previous weak law of large numbers

Applicable under power-like tail behavior with exponent -α

Abstract

We continue the line of research of random walks with barrier initiated by Iksanov and M{\"o}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-\alpha$, $\alpha\in(0,1)$, we prove that the number $V_n$ of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for $V_n$ proved in Iksanov and Negadailov (2008).