Abstract

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 Villaseñor [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, Rempała and Wesołowski [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 and Su [9], we have that if has a medium tail and , then , which implies that with probability one is finite at most. Thus, we can redefine if .

2. Main Result

Now we are ready to state our main results.

Theorem 2.1. Let be a sequence of positive i.i.d. random variables with a continuous distribution function , and . Assume that has a medium tail. Then, one has where is a standard Wiener process.

In particular, when we take , it yields the following corollary.

Corollary 2.2. Under the assumptions of Theorem 2.1, one has as , where is a standard normal random variable.

Remark 2.3. Since is a normal random variable with Corollary 2.2 follows from Theorem 2.1 immediately.

3. Proof of Theorem 2.1

In this section, we will give the proof of Theorem 2.1. In the sequel, let denote a positive constant which may take different values in different appearances and mean the largest integer .

Note that via Remark 1 of Hu and Su [9], we have . It follows that for any , there exists a positive integer such that Consequently, there exist two sequences and such that The strong law of large numbers also implies that there exists a sequence such that Here and in the sequel, we take , and it yields Then, it leads to and . By using the expansion of the logarithm , where depends on , we have that where () are (0-1)-valued and .

Also, we can rewrite as Observe that, for any fixed , it is easy to obtain by noting that .

And if , then we have as . If , then . Denote and, by observing that , then we can obtain For any , by the Markov’s inequality, we have Then, . To obtain this result, we need the following fact: Indeed, Now, we choose two constants and such that . Hence, in view of the strong law of large numbers, we have for large enough which together with (3.14) implies that as . Furthermore, in view of the strong law of large numbers again, we obtain as , where . For , by noting that , as (see Hu and Su [9]), thus we can easily get as . Then, for any , there exists a positive integer such that Consequently, coupled with (3.18), we have Clearly, to show , as , it is sufficient to prove Indeed, combined with (3.17), we only need to show As a matter of fact, by the definitions of and , we have In view of the fact , we can get from Hu and Su [9] that and thus it suffices to prove Actually, for all , and large enough, we can have that Observe that if has a medium tail, then we have by noting that [9], where is the limit defined in Section 1. Thus it follows as . Further, by the Markov’s inequality and the bounded property of from Hu and Su [9], we have and, hence, the proof of (3.22) is terminated. Thus follows. Finally, in order to complete the proof, it is sufficient to show that and, coupled with (3.21), we only need to prove Let It is obvious that Note that and then, for any , by the Cauchy-Schwarz inequality and (3.17), it follows that Furthermore, we can obtain Therefore, uniformly for , we have where . Notice that is a continuous mapping on the space . Thus, using the continuous mapping theorem (c.f., Theorem 2.7 of Billingsley [10]), it follows that Hence, (3.32), (3.34), and (3.37) coupled with Theorem 3.2 of Billingsley [10] lead to (3.30). The proof is now completed.