From Random Walks to Random Leaps: Generalizing Classic Markov Chains for Big Data Applications

TL;DR

This paper introduces the concept of random leaps, a generalization of random walks with larger step sizes, providing analytical tools for their behavior, especially in big data contexts, and exploring their properties and variants.

Contribution

It develops closed-form expressions for key quantities of random leaps, extending classical results, and analyzes the behavior of reflecting random leaps including stationary distributions.

Findings

Closed-form formulas for first passage times and absorption probabilities.

Random leaps generalize classical random walks with broader step sizes.

Reflecting random leaps can have stationary distributions and exhibit complex recurrence behavior.

Abstract

Simple random walks are a basic staple of the foundation of probability theory and form the building block of many useful and complex stochastic processes. In this paper we study a natural generalization of the random walk to a process in which the allowed step sizes take values in the set $\{\pm1,\pm2,\ldots,\pm k\}$, a process we call a random leap. The need to analyze such models arises naturally in modern-day data science and so-called "big data" applications. We provide closed-form expressions for quantities associated with first passage times and absorption events of random leaps. These expressions are formulated in terms of the roots of the characteristic polynomial of a certain recurrence relation associated with the transition probabilities. Our analysis shows that the expressions for absorption probabilities for the classical simple random walk are a special case of a universal result that is very elegant. We also consider an important variant of a random leap: the reflecting random leap. We demonstrate that the reflecting random leap exhibits more interesting behavior in regard to the existence of a stationary distribution and properties thereof. Questions relating to recurrence/transience are also addressed, as well as an application of the random leap.