# Functional Erd\H{o}s-R\'enyi laws for L\'evy processes

## TL;DR

This paper establishes functional Erdős-Rényi laws for Lévy processes, providing limit theorems for sets of functions related to their increments under large deviations principles.

## Contribution

It introduces new conditions linking large deviations principles to functional Erdős-Rényi laws for Lévy processes, expanding understanding of their sample path behavior.

## Key findings

- Derived conditions for convergence from large deviations principles.
- Established limit theorems under exponential moment assumptions.
- Connected large deviations to functional laws for Lévy processes.

## Abstract

In this paper we establish functional Erd\H{o}s-Renyi laws for L\'evy processes, i.e. limit theorems for sets of functions on [0,1] associated to their increments. First, we determine precise conditions under which, in a general framework, such a convergence is derived from a large deviations principle for probability measures induced by the sample paths of such a process. Then, by checking that these conditions are fulfilled, we obtain, under two usual assumptions on exponential moments, such limit theorems from well-known large deviations principles.