Probabilities of Large Deviations for Sums ofIndependent Random Variables with a Common Distribution Functionfrom the Domain of Attraction of an Asymmetric Stable Law

AbstractLet X1, X2, …, be a sequence of independent random variables with common distribution function F in the domain of attraction of a stable law and, for each n ≥ 1, let X1, n ≤ … ≤ Xn, n denote the order statistics based on the first n of these random variables. It is shown that sums of the middle portion of the order statistics of the form , where (kn)n ≥ 1 is a sequence of non-negative integers such that kn → ∞ and kn/n → 0 as n → ∞ at an appropriate rate, can be normalized and centred so that the law of the iterated logarithm holds. The method of proof is based on the almost sure properties of weighted uniform empirical processes.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2018/8968947 ◽

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.

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On the maximum of sums of random variables and the supremum functional for stable processes

Journal of Applied Probability ◽

10.2307/3212011 ◽

1969 ◽

Vol 6 (2) ◽

pp. 419-429 ◽

Cited By ~ 18

Author(s):

C.C. Heyde

Keyword(s):

Limit Theorem ◽

Stable Distribution ◽

Random Variables ◽

Domain Of Attraction ◽

Stable Processes ◽

Stable Law ◽

Sums Of Random Variables ◽

Symmetric Stable Distribution ◽

General Limit ◽

General Limit Theorem

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

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Some Functional Stable Limit Theorems

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-031-4 ◽

1973 ◽

Vol 16 (2) ◽

pp. 173-177 ◽

Cited By ~ 1

Author(s):

D. R. Beuerman

Keyword(s):

Weak Convergence ◽

Limit Theorems ◽

Stable Process ◽

Random Variables ◽

Domain Of Attraction ◽

Stable Processes ◽

Stable Limit ◽

Stable Law ◽

Partial Sum Process ◽

Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

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