A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law

Abstract

Let X, X 1 , X 2 ,... be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution. A universal result in almost sure limit theorem for the self-normalized partial sums S n /V n is established, whereKeywords: domain of attraction of the normal law, self-normalized partial sums, almost sure central limit theorem

An almost sure central limit theorem for self-normalized weighted sums of the φ mixing random variables

“…In this paper, an almost sure central limit theorem is obtained for self-normalized weighted sums of the φ mixing random variables. Our results extend and give substantial improvement for the result obtained by Zhang [12] and our results also extend the earlier work on almost sure central limit theorem such as Wu [13].…”

Almost sure central limit theorem for self-normalized partial sums of negatively associated random variables

“…The following four lemmas below play an important role in the proof of Theorem 1.1. Lemma 2.1 is due to Joag-Dev and Proschan [19], Lemma 2.2 has been stated by Su et al [21], Lemma 2.3 has been established by Wu [17], and Lemma 2.4 is of our authorship; due to its length, the proof of Lemma 2.4 is given in Appendix. Lemma 2.1.…”

“…The past decade has witnessed a significant development in the field of limit theorems for the self-normalized sum S n /V n . We refer to Bentkus and Gótze [11] for the Berry-Esseen bound, Gine et al [12] for the asymptotic normality, Hu et al [13] for the Cramer type moderate deviations, Csörgo et al [14] for the Donsker's theorem, Huang and Pang [15], Zhang and Yang [16] and Wu [17] for the almost sure central limit theorems. In addition, Wu [17] proved the ASCLT for the self-normalized partial sums that reads as follows: Let {X, X n } n∈N be a sequence of i.i.d.…”

An extension of almost sure central limit theorem for the maximum of stationary Gaussian random fields

“…Similarly to the proof of Lemma 2.2 inWu (2012a), we can prove Lemma 3.3.Proof of Theorem 2.1. By the terminology of summation procedures (see e.g Chandrasekharan and Minakshisundaram (1952),.…”