# Ladder Chains: A Variation of Random Walks

## TL;DR

This paper introduces ladder chains, a new variation of random walks, and extends classical concepts like ruin probability and recurrence, providing analytical solutions and proving recurrence in a critical case.

## Contribution

It defines ladder chains and generalizes key random walk concepts, offering new analytical methods and results for this variation.

## Key findings

- Derived difference equations for ruin probability and hitting time.
- Proved recurrence in the critical case p=√2−1.
- Generalized approaches to other ladder chains.

## Abstract

The authors propose a new variation of random walks called ladder chains $L(r,s,p)$. We extend concepts such as ruin probability, hitting time, transience and recurrence of random walks to ladder chain. Take $L(2,2,p)$ for instance, we find the linear difference equations that the ruin probability and the hitting time satisfy. We also prove the recurrence of a critical case $(p=\sqrt{2}-1)$. All approaches of these results can be generalized to solve similar problems for other ladder chains.