Almost sure and weak invariance principles for random variables attracted by a stable law

1989 ◽  
Vol 83 (3) ◽  
pp. 331-353 ◽  
Author(s):  
I. Berkes ◽  
H. Dehling
Keyword(s):  
Random Variables ◽  
Stable Law ◽  
Invariance Principles ◽  
Weak Invariance

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

1969 ◽  
Vol 6 (2) ◽  
pp. 419-429 ◽  
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.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
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.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Law of the iterated logarithm for self-normalized sums and their increments

2006 ◽  
Vol 43 (1) ◽  
pp. 79-114
Author(s):  
Han-Ying Liang ◽  
Jong-Il Baek ◽  
Josef Steinebach
Keyword(s):  
Random Variables ◽  
Domain Of Attraction ◽  
Weighted Sums ◽  
Partial Sums ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Independent And Identically Distributed

Let X1, X2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses Mn=max 1?i?n|Xi| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

Limit Theorems for an Extended Coupon Collector's Problem and for Successive Subsampling with Varying Probabilites*

1980 ◽  
Vol 29 (3-4) ◽  
pp. 113-132 ◽  
Author(s):  
Pranab Kumar Sen
Keyword(s):  
Asymptotic Normality ◽  
Limit Theorems ◽  
Finite Population ◽  
Waiting Times ◽  
Invariance Principles ◽  
Weak Invariance ◽  
Coupon Collector’S Problem ◽  
Asymptotic Distribution Theory ◽  
Multistage Sampling ◽  
Coupon Collector's Problem

Asymptotic normality as well as some weak invariance principles for bonus sums and waiting times in an extended coupon collector's problem are considered and incorporated in the study of the asymptotic distribution theory of estimators of (finite) population totals in successive sub-sampling (or multistage sampling) with varying probabilities (without replacement). Some applications of these theorems are also considered.

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