How linear reinforcement affects Donsker's Theorem for empirical processes

TL;DR

This paper investigates how linear reinforcement in a specific algorithm influences the convergence of empirical processes, showing that the classical Donsker theorem does not hold in general and requires modifications depending on the reinforcement probability.

Contribution

It demonstrates the impact of linear reinforcement on the Donsker theorem for empirical processes, revealing the need for rescaling and characterizing the limiting processes.

Findings

Convergence to a Brownian bridge only up to a constant factor when p<1/2.

Additional rescaling needed for p>1/2, leading to a limit with exchangeable increments.

Glivenko-Cantelli theorem remains valid despite reinforcement.

Abstract

A reinforcement algorithm introduced by H.A. Simon \cite{Simon} produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability $p\in(0,1)$, $\hat U_{n+1}$ is sampled uniformly from $\hat U_1, \ldots, \hat U_n$, and with complementary probability $1-p$, $\hat U_{n+1}$ is a new independent uniform variable. The Glivenko-Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $p<1/2$, and that a further rescaling is needed when $p>1/2$ and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.