A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process. 1. Introduction Let be a sequence of random variables and define the partial sum and for , where . In the past years, the asymptotic behaviors of the products of various random variables have been widely studied. Arnold and Villase？or [1] considered sums of records and obtained the following form of the central limit theorem (CLT) for independent and identically distributed (i.i.d.) exponential random variables with the mean equal to one, Here and in the sequel, is a standard normal random variable, and ( ) stands for convergence in distribution (in probability, almost surely). Observe that, via the Stirling formula, the relation (1.1) can be equivalently stated as In particular, Rempa？a and Weso？owski [2] removed the condition that the distribution is exponential and showed the asymptotic behavior of products of partial sums holds for any sequence of i.i.d. positive random variables. Namely, they proved the following theorem. Theorem A. Let be a sequence of i.i.d. positive square integrable random variables with and the coefficient of variation . Then, one has Recently, the above result was extended by Qi [3], who showed that whenever is in the domain of attraction of a stable law with index , there exists a numerical sequence (for , it can be taken as ) such that as , where . Furthermore, Zhang and Huang [4] extended Theorem A to the invariance principle. In this paper, we aim to study the weak invariance principle for self-normalized products of trimmed sums of i.i.d. sequences. Before stating our main results, we need to introduce some necessary notions. Let be a sequence of i.i.d. random variables with a continuous distribution function . Assume that the right extremity of satisfies and the limiting tail quotient exists, where . Then, the above limit is for some , and or is said to have a thick tail if , a medium tail if , and a thin tail if . Denote . For a fixed constant , we say is a near-maximum if and only if , and the number of near-maxima is These concepts were first introduced by Pakes and Steutel [5], and their limit properties have been widely studied by Pakes and Steutel [5], Pakes and Li [6], Li [7], Pakes [8], and Hu and Su [9]. Now, set where which are the sum of near-maxima and the trimmed sum, respectively. From Remark 1 of Hu