A note on the almost sure limit theorem for the product of partial sums

Abstract

We present an almost sure limit theorem for the product of the partial sums of i.i.d. positive random variables. We also prove a corresponding almost sure limit theorem for a triangular array.

Almost sure functional limit theorem for the product of partial sums

“…The goal of this paper is to obtain an almost sure version of the above invariance principle which can also be a functional version of the almost sure limit theorem obtained by Gonchigdanzan and Rempa la [3]. Here is the result: …”

Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima

“…ln t/˛for some˛ ˇ, whereˇis such as in (9), and that the conditions in (6)- (7) are satisfied. Then, the claim of Theorem 2.1 holds true with u n WD u n .x/ D a n x C b n and D e x , i.e., for any x 2 R, Proof.…”

“…Suppose that .X i / is a stationary, standard normal sequence satisfying (9) and that: for any fixed n 2 N, the random vector .X 1 ; :::; X n / has the Gumbel copula C ‰ with a generator of the form ‰ .t/ D . ln t/˛for some˛ ˇ, whereˇis such as in (9), and that the conditions in (6)- (7) are satisfied.…”

“…The topics pertaining to the ASCLTs have attracted an immense attention since the publication of the two above mentioned papers and a great deal of works devoted to the proofs of (1) for various classes of functions f n and random sequences .X i / have appeared throughout the last two decades or so. We cite in this context the articles by: Berkes and Csáki [3], Chen and Lin [4], Cheng et al [5], Csáki and Gonchigdanzan [6], Dudziński [7], Dudziński and Górka [8], Gonchigdanzan and Rempała [9], Ho and Hsing [10], Lacey and Philipp [11], Matuła [12], Mielniczuk [13], Peligrad and Shao [14], Stadtmüller [15], and Zhao et al [16], among others. Functions f n included different kinds of functions of r.v.…”

“…'s, e.g. : partial sums (see: [3], [7], [11]- [14]), products of partial sums (see [9]), maxima (see: [3]- [6]), extreme order statistics (see [15]), maxima of sums (see: [3], [8]), and -jointly -maxima and sums as well (see [16]). It is worth noting that not only the indicator functions need to be considered with regard to this issue -see, e.g., Fazekas and Rychlik [17]; we discuss the functional almost sure central limit theorem in this case.…”