An almost sure limit theorem for random sums of independent random variables in the domain of attraction of a semistable law

2009 ◽  
Vol 53 (11) ◽  
pp. 74-76
Author(s):  
A. N. Chuprunov ◽  
L. P. Terekhova
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Random Sums ◽  
Almost Sure Limit Theorem ◽  
Semistable Law

scholarly journals Almost Sure Central Limit Theory for Self-Normalized Products of Sums of Partial Sums

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Qunying Wu
Keyword(s):  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Domain Of Attraction ◽  
The Self ◽  
Partial Sums ◽  
Limit Theory ◽  
Almost Sure Limit Theorem ◽  
Products Of Sums ◽  
Normal Law

LetX,X1,X2,…be a sequence of independent and identically distributed random variables in the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized products of sums of partial sums is established.

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.

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