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 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.

scholarly journals A local limit theorem for random walks in random scenery and on randomly oriented lattices

2011 ◽  
Vol 39 (6) ◽  
pp. 2079-2118 ◽  
Author(s):  
Fabienne Castell ◽  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Bruno Schapira
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Random Scenery

An analog of the two-dimensional local limit theorem

1978 ◽  
Vol 29 (4) ◽  
pp. 415-421
Author(s):  
V. A. Peten'ko ◽  
Yu. P. Studnev
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional

A large deviation local limit theorem

1989 ◽  
Vol 105 (3) ◽  
pp. 575-577 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Deviation Result

The following elegant one-sided large deviation result is given by S. V. Nagaev in [2].Theorem 0. Suppose that {Sn,n ≤ 0} is a random walk whose increments Xi are independent copies of X, where(X) = 0 andPr{X > x} ̃ x−αL(x) as x→ + ∞,and where 1 < α < ∞ and L is slowly varying at ∞. Then for any ε > 0 and uniformly in x ≥ εnPr{Sn > x} ̃ n Pr{X > x} as n→∞.It is the purpose of this note to point out that for lattice-valued random walks there is an analogous local limit theorem.

scholarly journals A local limit theorem for a sequence of interval transformations

1985 ◽  
Vol 5 (2) ◽  
pp. 185-201 ◽  
Author(s):  
P. Calderoni ◽  
M. Campanino ◽  
D. Capocaccia
Keyword(s):  
Limit Theorem ◽  
Characteristic Function ◽  
Open Subset ◽  
Local Limit Theorem ◽  
Fractional Part ◽  
Local Limit ◽  
Real Eigenvalue ◽  
Two Dimensional ◽  
Dense Open Subset ◽  
Dimensional Torus

AbstractLet λ > 1 be a real eigenvalue of an automorphism of the two dimensional torus. We prove that for a dense, open subset of intervals the sequence where {x} denotes the fractional part of x and χ[a, b] the characteristic function of [a, b], satisfies the local limit theorem with respect to Lebesgue measure on [0, 1].

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