A large-deviation result for the range of random walk and for the Wiener sausage

2001 ◽  
Vol 120 (2) ◽  
pp. 183-208 ◽  
Author(s):  
Yuji Hamana ◽  
Harry Kesten
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Wiener Sausage ◽  
Large Deviation Result ◽  
Range Of Random Walk

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 Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

scholarly journals Approximation of bounds on mixed-level orthogonal arrays

2011 ◽  
Vol 43 (02) ◽  
pp. 399-421
Author(s):  
Ali Devin Sezer ◽  
Ferruh Özbudak
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Recursive Algorithm ◽  
Markov Property ◽  
Orthogonal Arrays ◽  
Recursive Formula ◽  
Limit Problem ◽  
Simulation Algorithm ◽  
Bellman Equations ◽  
Combinatorial Representations

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

LARGE DEVIATION ASYMPTOTICS FOR RANDOM-WALK TYPE PERTURBATIONS

2007 ◽  
Vol 07 (01) ◽  
pp. 75-89
Author(s):  
ZHIHUI YANG
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
White Noise ◽  
Wiener Process ◽  
Stochastic Resonance ◽  
Large Deviation ◽  
Correction Term ◽  
Exit Problem ◽  
Wiener Processes ◽  
Noise Type

Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

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