Constructions of majorizing measures Bernoulli processes and cotype

M. Talagrand

doi:10.1007/bf01896658

Constructions of majorizing measures Bernoulli processes and cotype

Geometric and Functional Analysis ◽

10.1007/bf01896658 ◽

1994 ◽

Vol 4 (6) ◽

pp. 660-717 ◽

Cited By ~ 8

Author(s):

M. Talagrand

Keyword(s):

Majorizing Measures ◽

Bernoulli Processes

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Bernoulli Processes

The Rademacher System in Function Spaces ◽

10.1007/978-3-030-47890-2_10 ◽

2020 ◽

pp. 291-310

Author(s):

Sergey V. Astashkin

Keyword(s):

Bernoulli Processes

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How linear reinforcement affects Donsker’s theorem for empirical processes

Probability Theory and Related Fields ◽

10.1007/s00440-020-01001-9 ◽

2020 ◽

Vol 178 (3-4) ◽

pp. 1173-1192 ◽

Cited By ~ 1

Author(s):

Jean Bertoin

Keyword(s):

Random Walk ◽

Random Walks ◽

Long Range ◽

Limit Theorems ◽

Brownian Bridge ◽

Empirical Processes ◽

Constant Factor ◽

Empirical Measure ◽

Reinforced Random Walks ◽

Bernoulli Processes

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 $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ 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 $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ 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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