# Independence times for iid sequences, random walks and L\'evy processes

## TL;DR

This paper characterizes and provides conditions for random times in iid sequences, random walks, and Lévy processes where the future process is independent of the past, enhancing understanding of independence times in stochastic processes.

## Contribution

It offers set-theoretic characterizations and conditions for independence times in iid sequences, random walks, and Lévy processes, including partial necessary and sufficient criteria.

## Key findings

- Characterization of independence times in discrete-time sequences
- Conditions for independence of future increments in Lévy processes
- Partial necessary conditions for independence times in Lévy processes

## Abstract

For a sequence in discrete time having stationary independent values (respectively, random walk) $X$, those random times $R$ of $X$ are characterized set-theoretically, for which the strict post-$R$ sequence (respectively, the process of the increments of $X$ after $R$) is independent of the history up to $R$. For a L\'evy process $X$ and a random time $R$ of $X$, reasonably useful sufficient conditions and a partial necessary condition on $R$ are given, for the process of the increments of $X$ after $R$ to be independent of the history up to $R$.