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.

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 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 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.

scholarly journals Limit theorems for isotropic random walks on triangle buildings

2002 ◽  
Vol 73 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Marc Lindlbauer ◽  
Michael Voit
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional ◽  
Littlewood Polynomials ◽  
Large Numbers ◽  
Moment Functions ◽  
Vertex Sets ◽  
Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

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 } .

On entrance times of a homogeneous N-dimensional random walk: an identity

1988 ◽  
Vol 25 (A) ◽  
pp. 321-333 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Performance Evaluation ◽  
Random Walk ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Mathematical Problem ◽  
Lattice Points ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Computer Performance ◽  
Hilbert Type

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

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