scholarly journals On the number of zero increments of random walks with a barrier

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Alex Iksanov ◽  
Pavlo Negadajlov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Regular Variation ◽  
Law Of Large Numbers ◽  
Large Numbers ◽  
International Audience

International audience Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as $n \to \infty$) behaviour of $V_n$ the number of zero increments before the absorption in a random walk with the barrier $n$. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for $V_n$ under a regular variation assumption.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (4) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

scholarly journals DYNAMIC QUANTUM BERNOULLI RANDOM WALKS

2008 ◽  
Vol 11 (02) ◽  
pp. 213-229
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
RENÉ SCHOTT
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Numbers

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (04) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z d . The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

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 } .

scholarly journals Excursions of a Random Walk Related to the Strong Law of Large Numbers

1998 ◽  
Vol 28 (2) ◽  
pp. 595-606
Author(s):  
Travis Lee ◽  
Max Minzner ◽  
Evan Fisher
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers

Records from stationary observations subject to a random trend

2015 ◽  
Vol 47 (04) ◽  
pp. 1175-1189 ◽  
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Random Walk ◽  
Asymptotic Normality ◽  
Strong Convergence ◽  
Ergodic Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Convergence Result ◽  
Stationary Sequences ◽  
Stochastic Trend ◽  
Large Numbers

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Y n = X n + T n , n ≥ 1, where (X n ) n ∈ Z is a stationary ergodic sequence of random variables and (T n ) n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

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