How linear reinforcement affects Donsker's Theorem for empirical processes
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A reinforcement algorithm introduced by H.A. Simon <cit.> produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability p∈(0,1), Û_n+1 is sampled uniformly from Û_1, ..., Û_n, and with complementary probability 1-p, Û_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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