Complete convergence for arrays of ratios of order statistics

Abstract Let {Xn,k, 1 ≤ k ≤ mn, n ≥ 1} be an array of independent random variables from the Pareto distribution. Let Xn(k) be the kth largest order statistic from the nth row of the array and set Rn,in,jn = Xn(jn)/Xn(in) where jn < in. The aim of this paper is to study the complete convergence of the ratios {Rn,in,jn, n ≥ 1}.


Introduction
Let {Xn , n ≥ } be a sequence of independent and identically distributed random variables and let X n(n) ≤ X n(n− ) ≤ · · · ≤ X n( ) be the order statistics. During the past many years, the in uence of order statistics has attracted considerable attention. This topic is relevant in various practical situations like in (re)insurance applications when a signi cant proportion of the sum of claims is consumed by a small number of claims (or even by a single claim) due to earthquakes, oods, hurricanes, terrorism, etc. Numerous authors gave necessary and/or su cient conditions for the convergence of certain ratios of extreme terms (order statistics and terms of maximum modulus) and sums. Smid and Stam [1] proved that the successive quotients of the order statistics in decreasing order are asymptotically independent with some distribution functions, under certain conditions. O'Brien [2] studied the ratio variate X n( ) /Sn, where Sn = X + · · · + Xn, which is a quantity arising in the analysis of process speedup and the performance of scheduling. Balakrishnan and Stepanov [3] discussed the asymptotic properties of ratios X n(j) /X n(i) for i ≠ j. Furthermore, there are many scholars who have studied the more-re ned properties for some speci c distributions. For example, Malik and Trudel [4] found the distribution of the quotient of and two order statistics from the Pareto, Power and Weibull distributions, by using the Mellin transform technique.
In the present paper, we want to study some limit theorems for the order statistics from the Pareto distribution. Pareto distribution describes the income of individuals, and since its introduction, it has found wide applications in many elds of studies such as economics, insurance premium, population distributions, stock market analysis, and queuing theory. Johnson et al. [5] discussed some potential applications of this distribution in di erent subject elds. Mann et al. [6] studied di erent estimation procedures for the unknown parameters of the Pareto distribution. Miao et al. [7] establish two large deviations for the Pareto distribution, and discussed the maxima of sums of the two-tailed Pareto random variables.
Let X be a random variable with the Pareto distribution, i.e., the probability density function of X is where xm is the (necessarily positive) minimum possible value of X, and k is a positive parameter. In the present paper, we consider an array of independent random variables {X n,k , ≤ k ≤ mn , n ≥ } with density where pn > . Let X n(kn) be the knth largest order statistic from each row of the array. Hence we have X n(mn) ≤ X n(mn− ) ≤ · · · ≤ X n( ) ≤ X n( ) .
Next let us de ne the ratios of the order statistics Rn := R n,in ,jn = X n(jn) X n(in) , jn < in which implies that Rn ≥ . It is not di cult to check that the density of Rn is It is obvious that the density of Rn is free of mn. Adler [8][9][10] studied the limit behaviors for the weighted sums of the ratios. In [9], Adler assumed that the parameters in , jn were xed. In [8], Adler was allowed to let mn and in grow, but jn was xed. The paper [10] was a natural extension of Adler [8,9] and allow all subscripts to grow, but the distance between in and jn was xed, i.e., ∆ := in − jn was xed. Adler [10] obtained the following results. The rst theorem establishes an unusual strong law where ∆ = .
or, if one wishes where naturally, if ∆ = we have ∆− k= k = .
Theorem 1.4. [10] If pn jn = , jn ∼ (log n) a , where < a < , ∆ = and α > − , then There is some literature concerning the limit theorems, for example, Miao et al. [11,12] established some limit theorems for the ratios of order statistics from exponentials and uniform distribution. In the present paper, we are interested in the complete convergence for the ratios Rn. In order to prove the complete convergence, since the expectation of Rn dose not exists, we need to deal with the tailed term by using the results in Sung et al. [13]. Throughout the paper, we use the constant C to denote a generic real number that is not necessarily the same in each appearance, and we de ne log x = log(max{e, x}).

Complete convergence theorems
In this section, we consider the complete convergence theorems for the weighted sums of the ratios Rn. The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [14] as follows. A sequence {Un , n ≥ } is said to converge completely to a constant C if ∞ n= P(|Un − C| > ε) < ∞, for all ε > .
By using Borel-Cantelli lemma, this implies that Un → C almost surely.

. Several lemmas
Before giving the main results, we need to recall the following some lemmas. Suppose that for every r > and some ε > : Lemma 2.2. [10] For ∆ > and pn jn = , we have where "an ∼ bn" denotes limn→∞ an /bn = .

. Main results
In the subsection, we give the complete convergence of the weighted sums with di erent forms. The distance between in and jn is xed, i.e., ∆ is xed. We consider the most interesting case and assume that pn jn = .

Remark 2.1. The convergence of the series in (2.3) holds trivially for any
where cn = nj ∆− n (log n) .
Remark 2.2. The conclusion of Theorem 2.2 can also be obtained when jn ∼ log n and cn = n(log n) ∆+ .

. Proofs of main results
Before giving the proofs, we want to show the outlines of the proofs. For any k, we partition k n= tn Rn − ERn I( ≤ Rn ≤ cn) into the following two terms: where tn are our weights. Next we discuss separately the complete convergence of the above terms. The complete convergence of the rst term can be veri ed by the di erent density of Rn, Lemma 2.2 and Kolmogorov's inequality. Since the expectation of Rn dose not exists, we will deal with the tailed term by using the results in Sung et al. [13]. Therefore, the complete convergence of the second term can be obtained by Lemma 2.1.
Proof of Theorem 2.1. Firstly we partition k n= ((log n) α /n) Rn − ERn I( ≤ Rn ≤ cn) into the following two terms: Note that the density for the ratio of our adjacent order statistics, that is, ∆ = , is then it is not di cult to see that (2.9) Here we use the fact Proof of Theorem 2.2. As the proof of Theorem 2.1, we partition k n= ((log n) α /(nj ∆− n ))(Rn−ERn I( ≤ Rn ≤ cn)) into the following two terms: For the case ∆ ≥ , pn jn = and by Lemma 2.2, the density for the ratio of our adjacent order statistics is Then it follows that By the Kolmogorov's inequality, we have Therefore, the desired result can be obtained by (2.18) and (2.22).
Proof of Theorem 2.3. As the proof of Theorem 2.1, we partition k n= ((log n) α /n)(Rn − ERn I( ≤ Rn ≤ cn)) into the following two terms: For the case ∆ = , pn jn = and by Lemma 2.2, the density for the ratio of our order statistics is Then it follows that (2.29) Therefore, the desired result can be obtained by (2.25) and (2.29).

Corollaries and examples
In this section, we investigate some corollaries and examples for the complete convergence theorems in Section 2. Adler [10] examined strong laws involving weighted sums of {Rn , n ≥ }, and get some unusual limit theorems. Our corollaries show the complete convergence about them. And then, we search some examples to enhance the persuasive of the conclusion for speci c r N and b N , N ≥ . If jn ∼ log n, then Proof. For the case jn = o(log n) and cn = nj ∆− n (log n) , we have By checking directly the conditions in the proof of Theorem 2.2, the conclusion (3.5) can be obtained.
For the case jn ∼ log n, let cn = n(log n) ∆+ , then it is not di cult to check that c − /jn n → e − as n → ∞. Hence we get By checking the conditions in the proof of Theorem 2.2 where jn ∼ log n, the conclusion (3.6) can be obtained. If jn ∼ log n, then If jn ∼ log n, then Then the conclusion can be obtained.

Conclusion
The work examines some limit theorems for the order statistics from the Pareto distribution proposed in current work, and investigates the complete convergence for the ratios of the order statistics. In this paper, we rstly get the complete convergence of the weighted sums with di erent forms by discussing three di erent cases. Then we obtain some corollaries. Some examples are also given to support the conclusion. There are more relevant properties regarding order statistics that will be investigated by us in the future.