An invariance principle for conditioned recurrent random walk attracted to a stable law

1972 ◽  
Vol 21 (1) ◽  
pp. 45-64 ◽  
Author(s):  
Barry Belkin
Keyword(s):  
Random Walk ◽  
Invariance Principle ◽  
Stable Law ◽  
Recurrent Random Walk

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.

On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks

2008 ◽  
Vol 28 (2) ◽  
pp. 423-446 ◽  
Author(s):  
Y. GUIVARC’H ◽  
EMILE LE PAGE
Keyword(s):  
Random Walk ◽  
Spectral Properties ◽  
Markov Operator ◽  
Dominant Eigenvalue ◽  
Fractional Powers ◽  
Stable Law ◽  
Fourier Operator ◽  
Transfer Operators ◽  
Heisenberg Type ◽  
Normal Law

AbstractWe consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on $\mathbb {R}$, and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

Limiting behaviour of the distributions of the maxima of partial sums of certain random walks

1972 ◽  
Vol 9 (3) ◽  
pp. 572-579 ◽  
Author(s):  
D. J. Emery
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Asymptotic Property ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Hitting Times ◽  
Partial Sums ◽  
Symmetric Distributions ◽  
Stable Law ◽  
Limiting Behaviour

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.

scholarly journals Universal Spectra of Coherent Atoms in a Recurrent Random Walk

2009 ◽  
Vol 102 (15) ◽  
Author(s):  
R. Pugatch ◽  
O. Firstenberg ◽  
M. Shuker ◽  
N. Davidson
Keyword(s):  
Random Walk ◽  
Recurrent Random Walk
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