Functional Law of the Iterated Logarithm for I.I.D. Random Variables Attracted to a Completely Asymmetric Stable Law

1980 ◽  
Vol 24 (4) ◽  
pp. 833-838
Author(s):  
R. Vasudeva
Keyword(s):  
Random Variables ◽  
Stable Law ◽  
Iterated Logarithm

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.

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.

scholarly journals Chover-type laws of the iterated logarithm for weighted sums of ρ∗-mixing sequences

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-hui Cai
Keyword(s):  
Random Variables ◽  
Weighted Sums ◽  
Type Result ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Mixing Sequences

To derive a Baum-Katz-type result, we establish a Chover-type law of the iterated logarithm for the weighted sums of ρ∗-mixing and identically distributed random variables with a distribution in the domain of a stable law. Our result obtained not only generalizes the main results of Peng and Qi (2003) and Qi and Cheng (1996) to ρ∗-mixing sequences of random variables, but also improves them.

Laws of the iterated logarithm for sums of the middle portion of the sample

1987 ◽  
Vol 101 (2) ◽  
pp. 301-312 ◽  
Author(s):  
Erich Haeusler ◽  
David M. Mason
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Middle Portion ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Common Distribution ◽  
Common Distribution Function

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.

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

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