scholarly journals Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

2017 ◽  
Vol 127 (4) ◽  
pp. 1282-1296 ◽  
Author(s):  
Zhiqiang Gao
Keyword(s):  
Limit Theorem ◽  
Convergence Rate ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Integer Lattice ◽  
Branching Random Walks

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 Local limit theorem for the maximum of asymptotically stable random walks

2010 ◽  
Vol 152 (3-4) ◽  
pp. 407-424 ◽  
Author(s):  
Vitali Wachtel
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Asymptotically Stable

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

scholarly journals An improvement of convergence rate in the local limit theorem for integral-valued random variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tatpon Siripraparat ◽  
Kritsana Neammanee
Keyword(s):  
Limit Theorem ◽  
Convergence Rate ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Moment Condition ◽  
Formula One ◽  
Explicit Constant ◽  
Independent Integral

AbstractLet $X_{1}, X_{2}, \ldots , X_{n}$ X 1 , X 2 , … , X n be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ S n = ∑ j = 1 n X j . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ S n . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ O ( 1 n ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ C n . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ S n . Our constants are less complicated than before, and thus easy to use.

Sign in / Sign up

Export Citation Format

Share Document