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.

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

1969 ◽  
Vol 6 (02) ◽  
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 S 0= 0, S n = Σ i=1 n Xi , n ≧ 1, and Mn = max0 ≦ k ≦ n Sk . In the case where the Xi are such that Σ1 ∞ n −1Pr(Sn &gt; 0) &lt; ∞, we have lim n→∞M n = M which is finite with probability one, while in the case where Σ1 ∞ n −1Pr(Sn &lt; 0) &lt; ∞, 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 &lt; 0) &lt; ∞, Σ1 ∞ n −1Pr(Sn &gt; 0) &lt; ∞ (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.

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.

A simple proof of a random stable limit theorem

1970 ◽  
Vol 7 (2) ◽  
pp. 502-504 ◽  
Author(s):  
Stephen R. Kimbleton
Keyword(s):  
Limit Theorem ◽  
Elementary Proof ◽  
General Theorem ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Limit ◽  
Stable Law ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

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