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How linear reinforcement affects Donsker’s theorem for empirical processes


Bertoin, Jean (2020). How linear reinforcement affects Donsker’s theorem for empirical processes. Probability Theory and Related Fields, 178:1173-1192.

Abstract

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability p∈(0,1), U^n+1 is sampled uniformly from U^1,…,U^n, and with complementary probability 1−p, 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.

Abstract

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability p∈(0,1), U^n+1 is sampled uniformly from U^1,…,U^n, and with complementary probability 1−p, 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability, Analysis
Language:English
Date:18 September 2020
Deposited On:01 Oct 2020 09:54
Last Modified:19 Feb 2021 08:41
Publisher:Springer
ISSN:0178-8051
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1007/s00440-020-01001-9

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