On a Maximum of a transient random walk in random environment

1991 ◽  
Vol 35 (2) ◽  
pp. 205-215 ◽  
Author(s):  
V. I. Afanas’ev
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Transient Random Walk

On the non-recurrent random walk in a random environment

2018 ◽  
Vol 28 (3) ◽  
pp. 139-156 ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Environment ◽  
Recurrent Random Walk ◽  
High Level ◽  
Transient Random Walk

Abstract For weakly transient random walk in a random environment that tend at −∞ the limit theorem for the time of hitting a high level is proved.

scholarly journals Ballisticity conditions for random walk in random environment

2010 ◽  
Vol 150 (1-2) ◽  
pp. 61-75 ◽  
Author(s):  
A. Drewitz ◽  
A. F. Ramírez
Keyword(s):  
Random Walk ◽  
Random Environment

scholarly journals Random walk in a high density dynamic random environment

2014 ◽  
Vol 25 (4) ◽  
pp. 785-799 ◽  
Author(s):  
Frank den Hollander ◽  
Harry Kesten ◽  
Vladas Sidoravicius
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
High Density ◽  
Density Dynamic

scholarly journals Random walk in cooling random environment: ergodic limits and concentration inequalities

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Luca Avena ◽  
Yuki Chino ◽  
Conrado da Costa ◽  
Frank den Hollander
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Concentration Inequalities

scholarly journals Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$

2019 ◽  
Vol 33 (4) ◽  
pp. 2315-2336
Author(s):  
Inna M. Asymont ◽  
Dmitry Korshunov
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Acta Math ◽  
Local Times ◽  
List Type ◽  
Strong Law ◽  
Statistics And Probability ◽  
Large Numbers ◽  
Berkeley Symposium ◽  
Transient Random Walk

Abstract For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1$$ d ≥ 1 , we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x ∈ Z d f ( l ( n , x ) ) of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } . Particular cases are the number of visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; $$\alpha $$ α -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$ f ( i ) = i α ; sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$ f ( i ) = I { i = j } .

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