Displaying similar documents to “On Feller's criterion for the law of the iterated logarithm.”

Necessary and sufficient conditions for weak convergence of random sums of independent random variables

Andrzej Krajka, Zdzisław Rychlik (1993)

Commentationes Mathematicae Universitatis Carolinae

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Let { X n , n ≥ 1 } be a sequence of independent random variables such that E X n = a n , E ( X n - a n ) 2 = σ n 2 , n ≥ 1 . Let { N n , n ≥ 1 } be a sequence od positive integer-valued random variables. Let us put S N n = ∑ k = 1 N n X k , L n = ∑ k = 1 n a k , s n 2 = ∑ k = 1 n σ k 2 , n ≥ 1 . In this paper we present necessary and sufficient conditions for weak convergence of the sequence { ( S N n - L n ) / s n , n ≥ 1 } , as n → ∞ . The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences

Jérôme Dedecker, Sana Louhichi (2005)

ESAIM: Probability and Statistics

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We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the gaussian and the purely non-gaussian parts of the infinitely divisible limit....