A Class of Self-Interacting Processes with Applications to Games and Reinforced Random Walks

2010 ◽  
Vol 48 (7) ◽  
pp. 4707-4730 ◽  
Author(s):  
Michel Benaïm ◽  
Olivier Raimond
Keyword(s):  
Random Walks ◽  
Reinforced Random Walks

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
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.

scholarly journals Preferential duplication graphs

2010 ◽  
Vol 47 (02) ◽  
pp. 572-585 ◽  
Author(s):  
Netta Cohen ◽  
Jonathan Jordan ◽  
Margaritis Voliotis
Keyword(s):  
Random Walks ◽  
Stochastic Approximation ◽  
Random Graphs ◽  
Reinforced Random Walks ◽  
Special Case

We consider a preferential duplication model for growing random graphs, extending previous models of duplication graphs by selecting the vertex to be duplicated with probability proportional to its degree. We show that a special case of this model can be analysed using the same stochastic approximation as for vertex-reinforced random walks, and show that ‘trapping’ behaviour can occur, such that the descendants of a particular group of initial vertices come to dominate the graph.

scholarly journals Edge oriented reinforced random walks and RWRE

2002 ◽  
Vol 335 (11) ◽  
pp. 941-946 ◽  
Author(s):  
Nathanaël Enriquez ◽  
Christophe Sabot
Keyword(s):  
Random Walks ◽  
Reinforced Random Walks

The ‘magic formula’ for linearly edge-reinforced random walks

2008 ◽  
Vol 62 (3) ◽  
pp. 345-363
Author(s):  
Franz Merkl ◽  
Aniko Öry ◽  
Silke W. W. Rolles
Keyword(s):  
Random Walks ◽  
Magic Formula ◽  
Reinforced Random Walks
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