Complete convergence for arrays of ratios of order statistics

Yu Miao; Huanhuan Ma; Shoufang Xu; Andre Adler

doi:10.1515/math-2019-0035

Article Open Access

Complete convergence for arrays of ratios of order statistics

Yu Miao , Huanhuan Ma , Shoufang Xu and Andre Adler

Published/Copyright: June 6, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Let {X_n,k, 1 ≤ k ≤ m_n, n ≥ 1} be an array of independent random variables from the Pareto distribution. Let X_n(k) be the kth largest order statistic from the nth row of the array and set R_{n,i_n,j_n} = X_{n(j_n)}/X_{n(i_n)} where j_n < i_n. The aim of this paper is to study the complete convergence of the ratios {R_{n,i_n,j_n}, n ≥ 1}.

Keywords: Ratios; Pareto distribution; complete convergence

MSC 2010: 60F15; 62G30

1 Introduction

Let {X_n, n ≥ 1} be a sequence of independent and identically distributed random variables and let X_n(n) ≤ X_n(n–1) ≤ ⋯ ≤ X_n(1) be the order statistics. During the past many years, the influence of order statistics has attracted considerable attention. This topic is relevant in various practical situations like in (re)insurance applications when a significant proportion of the sum of claims is consumed by a small number of claims (or even by a single claim) due to earthquakes, floods, hurricanes, terrorism, etc. Numerous authors gave necessary and/or sufficient 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(1)/S_n, where S_n = X₁ + ⋯ + X_n, 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-refined properties for some specific 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 fields 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 different subject fields. Mann et al. [6] studied different 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

f(x)=kxmk/xk+1I(x≥xm)

where x_m 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, 1 ≤ k ≤ m_n, n ≥ 1} with density

f(x)=pnx−pn−1I(x≥1)

where p_n > 0. Let X_{n(k_n)} be the k_n th largest order statistic from each row of the array. Hence we have

Xn(mn)≤Xn(mn−1)≤⋯≤Xn(2)≤Xn(1).

Next let us define the ratios of the order statistics

Rn:=Rn,in,jn=Xn(jn)Xn(in), jn<in

which implies that R_n ≥ 1. It is not difficult to check that the density of R_n is

fRn(r)=pn(in−1)!(in−jn−1)!(jn−1)!r−pnjn−1(1−r−pn)in−jn−1I(r≥1).

It is obvious that the density of R_n is free of m_n. Adler [8, 9, 10] studied the limit behaviors for the weighted sums of the ratios. In [9], Adler assumed that the parameters i_n, j_n were fixed. In [8], Adler was allowed to let m_n and i_n grow, but j_n was fixed. The paper [10] was a natural extension of Adler [8, 9] and allow all subscripts to grow, but the distance between i_n and j_n was fixed, i.e., Δ := i_n – j_n was fixed. Adler [10] obtained the following results. The first theorem establishes an unusual strong law where Δ = 1.

Theorem 1.1

[10] If p_nj_n = 1, Δ = 1 and α > –2, then

limN→∞∑n=1N((log⁡n)α/n)Rn(log⁡N)α+2=1α+2, a.s.

The following results are to consider the case Δ > 1.

Theorem 1.2

[10] If p_nj_n = 1, j_n = o(log n), Δ ≥ 2 and α > –2, then

limN→∞∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2=1(α+2)(Δ−1)!, a.s.

Theorem 1.3

[10] If p_nj_n = 1, j_n ∼ log n, Δ ≥ 2 and α > –2, then

limN→∞∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2=γΔ(α+2)(Δ−1)!, a.s.

where

γΔ=∑k=1Δ−1Δ−1k(−1)k+1e−kk−∑k=2Δ−11k,

or, if one wishes

γΔ=1+∑k=1Δ−1Δ−1k(−1)k(1−e−k)k,

where naturally, if Δ = 2 we have ∑k=2Δ−11k=0. .

Theorem 1.4

[10] If p_nj_n = 1, j_n ∼ (log n)^a, where 1 < a < 2, Δ = 2 and α > –3, then

limN→∞∑n=1N((log⁡n)α/n)Rn(log⁡N)α+3=12(α+3), a.s.

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 R_n. In order to prove the complete convergence, since the expectation of R_n 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 define log x = log(max{e, x}).

2 Complete convergence theorems

In this section, we consider the complete convergence theorems for the weighted sums of the ratios R_n. The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [14] as follows. A sequence {U_n, n ≥ 1} is said to converge completely to a constant C if

∑n=1∞P(|Un−C|>ε)<∞, for all ε>0.

By using Borel-Cantelli lemma, this implies that U_n → C almost surely.

2.1 Several lemmas

Before giving the main results, we need to recall the following some lemmas.

Lemma 2.1

[13] Let {X_n,k, 1 ≤ k ≤ m_n, n ≥ 1} be an array of rowwise independent random variables and {a_n, n ≥ 1} a sequence of positive constants such that

∑n=1∞an=∞.

Suppose that for every r > 0 and some ε > 0:

∑n=1∞an∑i=1mnP(|Xn,i|>r)<∞,
there exists J ≥ 2 such that

∑n=1∞an∑i=1mnEXn,i2I(|Xn,i|≤ε)J<∞,
∑i=1mnEXn,iI(|Xn,i|≤ε)→0 as n→∞.

Then we have

∑n=1∞anP∑i=1mnXn,i>r<∞ for all r>0.

Lemma 2.2

[10] For Δ > 1 and p_nj_n = 1, we have

pn(in−1)!(in−jn−1)!(jn−1)!∼jnΔ−1(Δ−1)!

where “a_n ∼ b_n” denotes lim_n→∞ a_n/b_n = 1.

Lemma 2.3

[10] If 1 < a < 2, then

limx→∞1+3x−1log⁡x+xa−1[(exx3)−1/xa−1]x1−a=1/2.

2.2 Main results

In the subsection, we give the complete convergence of the weighted sums with different forms. The distance between i_n and j_n is fixed, i.e., Δ is fixed. We consider the most interesting case and assume that p_nj_n = 1. The growth of j_n can’t be very fast, so we need a logarithmic growth rate for j_n. The conclusion breaks into three cases for different forms.

Δ = 1 and α > –2;
j_n = o(log n), Δ ≥ 2 and α > –2;
j_n ∼ (log n)^a, Δ = 2 and α > –3, where 1 < a < 2.

Theorem 2.1

Let {b_N, N ≥ 1} and {r_N, N ≥ 1} be sequences of positive numbers satisfying

∑N=1∞bN=∞, (loglog⁡N)(loglog⁡N+log⁡rN)rNlog⁡N→0 (2.1)

and

∑N=1∞bNloglog⁡NrNlog⁡Nmax1,1rN<∞. (2.2)

If p_nj_n = 1, Δ = 1 and α > –2, then

∑N=1∞bNPmax1≤k≤N∑n=1k(log⁡n)αnRn−ERnI(1≤Rn≤cn)≥rN(log⁡N)α+2<∞ (2.3)

where c_n = n(log n)².

Remark 2.1

The convergence of the series in (2.3) holds trivially for any ∑N=1∞ b_N < ∞.

Theorem 2.2

Let {b_N, N ≥ 1} and {r_N, N ≥ 1} be sequences of positive numbers satisfying (2.1) and (2.1). If p_nj_n = 1, j_n = o(log n), Δ ≥ 2 and α > –2, then

∑N=1∞bNPmax1≤k≤N∑n=1k(log⁡n)αnjnΔ−1Rn−ERnI(1≤Rn≤cn)≥rN(log⁡N)α+2<∞ (2.4)

where c_n = njnΔ−1 (log n)².

Remark 2.2

The conclusion of Theorem 2.2 can also be obtained when j_n ∼ log n and c_n = n(log n)^Δ+1.

Theorem 2.3

Let {b_N, N ≥ 1} and {r_N, N ≥ 1} be sequences of positive numbers satisfying

∑N=1∞bN=∞, (loglog⁡N)(loglog⁡N+log⁡rN)rN(log⁡N)2−a→0 (2.5)

and

∑N=1∞bNloglog⁡NrN(log⁡N)2−amax1,1rN<∞ (2.6)

where 1 < a < 2. If p_nj_n = 1, j_n ∼ (log n)^a, Δ = 2 and α > –3, then

∑N=1∞bNPmax1≤k≤N∑n=1k(log⁡n)αnRn−ERnI(1≤Rn≤cn)≥rN(log⁡N)α+3<∞

where c_n = n(log n)³.

2.3 Proofs of main results

Before giving the proofs, we want to show the outlines of the proofs. For any k, we partition ∑n=1ktnRn−ERnI(1≤Rn≤cn) into the following two terms:

∑n=1ktnRn−ERnI(1≤Rn≤cn)=∑n=1ktn[RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn)]+∑n=1ktnRnI(Rn>cn)=:Ik+IIk, (2.7)

where t_n are our weights. Next we discuss separately the complete convergence of the above terms. The complete convergence of the first term can be verified by the different density of R_n, Lemma 2.2 and Kolmogorov’s inequality. Since the expectation of R_n 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 ∑n=1k ((log n)^α/n)(R_n – ER_nI(1 ≤ R_n ≤ c_n)) into the following two terms:

∑n=1k((log⁡n)α/n)Rn−ERnI(1≤Rn≤cn)=∑n=1k((log⁡n)α/n)[RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn)]+∑n=1k((log⁡n)α/n)RnI(Rn>cn) =:Ik+IIk. (2.8)

Note that the density for the ratio of our adjacent order statistics, that is, Δ = 1, is

fRn(x)=x−2I(x≥1),

then it is not difficult to see that

ERn2I(1≤Rn≤cn)=cn−1=n(log⁡n)2−1.

By denoting

R~n:=RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn),

and by the Kolmogorov’s inequality, we have

Pmax1≤k≤N∑n=1k((log⁡n)α/n)R~n>rN(log⁡N)α+2≤1rN2(log⁡N)2(α+2)Var∑n=1N((log⁡n)α/n)R~n≤∑n=1N((log⁡n)α/n)2rN2(log⁡N)2(α+2)Var[RnI(1≤Rn≤cn)]≤∑n=1N((log⁡n)α/n)2rN2(log⁡N)2(α+2)n(log⁡n)2≤loglog⁡NrN2log⁡N. (2.9)

Here we use the fact

∑n=1N(log⁡n)(2α+2)n(log⁡N)2(α+2)≤Clog⁡N, for α>−2,α≠−32Cloglog⁡Nlog⁡N, for α=−32, (2.10)

for all N large enough. From the condition (2.1), we can get

∑N=1∞bNPmax1≤k≤N|Ik|≥rN(log⁡N)α+2<∞. (2.11)

Next, we consider the complete convergence of the term II_k, from using Lemma 2.1. By denoting

XN,n:=((log⁡n)α/n)RnI(Rn>cn)rN(log⁡N)α+2,

then by the similar discussions in (2.10), for any r > 0, we have

∑n=1NPXN,n≥r≤∑n=1NPRn≥maxcn,nr⋅rN(log⁡N)α+2/(log⁡n)α ≤C∑n=1Nmax1n(log⁡n)2,(log⁡n)αnrN(log⁡N)α+2≤Cloglog⁡NrNlog⁡N

which implies

∑N=1∞bN∑n=1NPXN,n≥r<∞. (2.12)

Furthermore, for any ε > 0, we have

∑n=1NEXN,nIXN,n≤ε≤Cloglog⁡N+log⁡rNrN(log⁡N)α+2∑n=1N(log⁡n)αn≤C(loglog⁡N)(loglog⁡N+log⁡rN)rNlog⁡N→0 (2.13)

as N → ∞, since the condition (2.1). At last

∑n=1NEXN,n2IXN,n≤ε≤C∑n=1N(log⁡n)α/nrN(log⁡N)α+2≤Cloglog⁡NrNlog⁡N,

then we get

∑N=1∞bN∑n=1NEXN,n2IXN,n≤ε2<∞. (2.14)

By Lemma 2.1 and (2.12)-(2.14), we have

∑N=1∞bNPmax1≤k≤N|IIk|≥r⋅rN(log⁡N)α+2=∑N=1∞bNP((log⁡n)α/n)RnI(Rn>cn)rN(log⁡N)α+2≥r<∞. (2.15)

Therefore, the desired result can be obtained by (2.11) and (2.15).□

Proof of Theorem 2.2

As the proof of Theorem 2.1, we partition ∑n=1k((log⁡n)α/(njnΔ−1))(Rn−ERnI(1≤Rn≤cn)) into the following two terms:

∑n=1k((log⁡n)α/(njnΔ−1))(Rn−ERnI(1≤Rn≤cn))=∑n=1k((log⁡n)α/(njnΔ−1))[RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn)]+∑n=1k((log⁡n)α/(njnΔ−1))RnI(Rn>cn)=:Ik+IIk. (2.16)

For the case Δ ≥ 2, p_nj_n = 1 and by Lemma 2.2, the density for the ratio of our adjacent order statistics is

fRn(x)=pn(in−1)!(in−jn−1)!(jn−1)!x−pnjn−1(1−x−pn)in−jn−1I(x≥1)∼jnΔ−1(Δ−1)!x−2(1−x−1/jn)Δ−1I(x≥1).

Then it follows that

ERnI(1≤Rn≤cn)∼jnΔ−1(Δ−1)!∫1cnx−1(1−x−1/jn)Δ−1dx=jnΔ−1(Δ−1)!∫1cnx−1∑k=0Δ−1Δ−1k(−1)kx−k/jndx=jnΔ−1(Δ−1)!log⁡cn+∑k=1Δ−1Δ−1k(−1)k(k/jn)−1(1−cn−k/jn)∼jnΔ−1log⁡n(Δ−1)!

and

ERn2I(1≤Rn≤cn)≤CjnΔ−1cn=Cnjn2(Δ−1)(log⁡n)2.

By the Kolmogorov’s inequality, we have

Pmax1≤k≤N∑n=1k((log⁡n)α/(njnΔ−1))R~n>rN(log⁡N)α+2≤∑n=1N((log⁡n)α/(njnΔ−1))2rN2(log⁡N)2(α+2)Var[RnI(1≤Rn≤cn)]≤∑n=1N((log⁡n)α/(njnΔ−1))2rN2(log⁡N)2(α+2)njn2(Δ−1)(log⁡n)2≤loglog⁡NrN2log⁡N (2.17)

where

R~n:=RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn).

From the condition (2.1), we can get

∑N=1∞bNPmax1≤k≤N|Ik|≥rN(log⁡N)α+2<∞. (2.18)

Next, we consider the complete convergence of the term II_k, from using Lemma 2.1. By denoting

XN,n:=((log⁡n)α/(njnΔ−1))RnI(Rn>cn)rN(log⁡N)α+2,

then for any r > 0, we have

∑n=1NPXN,n≥r≤∑n=1NPRn≥maxcn,nr⋅rNjnΔ−1(log⁡N)α+2/(log⁡n)α ≤C∑n=1Nmax1n(log⁡n)2,(log⁡n)αnrN(log⁡N)α+2≤Cloglog⁡NrNlog⁡N

which implies

∑N=1∞bN∑n=1NPXN,n≥r<∞. (2.19)

Furthermore, for any ε > 0, we have

∑n=1NEXN,nIXN,n≤ε≤CrN(log⁡N)α+2∑n=1N(log⁡n)αnjnΔ−1jnΔ−1(loglog⁡N+log⁡rN)(Δ−1)!≤Cloglog⁡N+log⁡rNrN(log⁡N)α+2∑n=1N(log⁡n)αn≤C(loglog⁡N)(loglog⁡N+log⁡rN)rNlog⁡N→0 (2.20)

as N → ∞. At last, since

∑n=1NEXN,n2IXN,n2≤ε≤C∑n=1NjnΔ−1(Δ−1)!ε(log⁡n)αnjnΔ−1rN(log⁡N)α+2 ≤C∑n=1N(log⁡n)α/nrN(log⁡N)α+2 ≤Cloglog⁡NrNlog⁡N,

we get

∑N=1∞bN∑n=1NEXN,n2IXN,n≤ε2<∞. (2.21)

By Lemma 2.1 and (2.19)-(2.21), we have

∑N=1∞bNPmax1≤k≤N|IIk|≥r⋅rN(log⁡N)α+2=∑N=1∞bNP((log⁡n)α/(njnΔ−1))RnI(Rn>cn)rN(log⁡N)α+2≥r<∞. (2.22)

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 ∑n=1k((log⁡n)α/n)(Rn−ERnI(1≤Rn≤cn)) into the following two terms:

∑n=1k((log⁡n)α/n)(Rn−ERnI(1≤Rn≤cn))=∑n=1k((log⁡n)α/n)[RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn)]+∑n=1k((log⁡n)α/n)RnI(Rn>cn)=:Ik+IIk. (2.23)

For the case Δ = 2, p_nj_n = 1 and by Lemma 2.2, the density for the ratio of our order statistics is

fRn(x)=pn(in−1)!(in−jn−1)!(jn−1)!x−pnjn−1(1−x−pn)in−jn−1I(x≥1)∼jnx−2(1−x−1/jn)I(x≥1).

Then it follows that

ERn2I(1≤Rn≤cn)∼jn∫1cn(1−x−1/jn)dx≤Cjncn∼Cn(log⁡n)a+3.

and

Pmax1≤k≤N∑n=1k((log⁡n)α/n)R~n>rN(log⁡N)α+3≤∑n=1N((log⁡n)α/n)2rN2(log⁡N)2(α+3)Var[RnI(1≤Rn≤cn)]≤∑n=1N((log⁡n)α/n)2rN2(log⁡N)2(α+3)n(log⁡n)a+3≤Cloglog⁡NrN2(log⁡N)2−a (2.24)

where

R~n:=RnI(1≤Rn≤cn)−ERnI(1≤Rn≤cn).

From the condition (2.6), we can get

∑N=1∞bNPmax1≤k≤N|Ik|≥rN(log⁡N)α+3<∞. (2.25)

Next, we consider the complete convergence of the term II_k, from using Lemma 2.1. By denoting

XN,n:=((log⁡n)α/n)RnI(Rn>cn)rN(log⁡N)α+3,

then for any r > 0, we have

∑n=1NPXN,n≥r≤∑n=1NPRn≥maxcn,nr⋅rN(log⁡N)α+3/(log⁡n)α≤C∑n=1Nmaxjncn,(log⁡n)α+anrN(log⁡N)α+3≤C∑n=1Nmax1n(log⁡n)3−a,(log⁡n)α+anrN(log⁡N)α+3≤Cloglog⁡NrN(log⁡N)2−a

which implies

∑N=1∞bN∑n=1NPXN,n≥r<∞. (2.26)

Furthermore, for any ε > 0, we have

∑n=1NEXN,nIXN,n≤ε≤Cloglog⁡N+log⁡rNrN(log⁡N)α+3∑n=1N(log⁡n)α+an≤C(loglog⁡N)(loglog⁡N+log⁡rN)rN(log⁡N)2−a→0 (2.27)

as N → ∞, since the condition (2.5). Then

∑n=1NEXN,n2IXN,n≤ε≤C∑n=1N(log⁡n)α+a/nrN(log⁡N)α+3≤Cloglog⁡NrN(log⁡N)2−a,

and

∑N=1∞bN∑n=1NEXN,n2IXN,n≤ε2<∞. (2.28)

By Lemma 2.1 and (2.26)-(2.28), we get

∑N=1∞bNPmax1≤k≤N|IIk|≥r⋅rN(log⁡N)α+3=∑N=1∞bNP((log⁡n)α/n)RnI(Rn>cn)rN(log⁡N)α+3≥r<∞. (2.29)

Therefore, the desired result can be obtained by (2.25) and (2.29).□

3 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 {R_n, n ≥ 1}, 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 specific r_N and b_N, N ≥ 1.

Corollary 3.1

Let {b_N, N ≥ 1} and {r_N, N ≥ 1} satisfy the condition (2.1) and (2.1). If p_nj_n = 1, Δ = 1 and α > –2, then

∑N=1∞bNP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+2−1α+2≥rN<∞. (3.1)

Proof

We can check directly the conditions in the proof of Theorem 2.1 to get the corollary, since

ERnI(1≤Rn≤cn)=log⁡cn∼log⁡n

and

∑n=1N((log⁡n)α/n)ERnI(1≤Rn≤cn)(log⁡N)α+2∼1α+2.

□

Example 3.1

For p_nj_n = 1, Δ = 1 and α > –2, let r_N = N and bN=1(log⁡N)δ for any δ > 0. Then we can get

∑N=1∞1(log⁡N)δP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+2−1α+2≥N<∞. (3.2)

Example 3.2

For p_nj_n = 1, Δ = 1 and α > –2, let rN=1loglog⁡N and bN=1N(loglog⁡N)δ+4 for any δ > 0. Then we can get

∑N=1∞1N(loglog⁡N)δ+4P∑n=1N((log⁡n)α/n)Rn(log⁡N)α+2−1α+2≥1loglog⁡N<∞. (3.3)

Example 3.3

If p_nj_n = 1, Δ = 1 and α > –2, let r_N = r be a constant and bN=1N(log⁡N)δ, where 0 < δ≤ 1. Then for r > 0, we have

∑N=1∞1N(log⁡N)δP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+2−1α+2≥r<∞. (3.4)

Corollary 3.2

For p_nj_n = 1, Δ ≥ 2 and α > –2, let {b_N, N ≥ 1} and {r_N, N ≥ 1} satisfy the condition (2.1) and (2.2). If j_n = o(log n), then

∑N=1∞bNP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−1(α+2)(Δ−1)!≥rN<∞. (3.5)

If j_n ∼ log n, then

∑N=1∞bNP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−γΔ(α+2)(Δ−1)!≥rN<∞ (3.6)

where

γΔ=1+∑k=1Δ−1Δ−1k(−1)k(1−e−k)k.

Proof

For the case j_n = o(log n) and cn=njnΔ−1(log⁡n)2, we have

ERnI(1≤Rn≤cn)∼jnΔ−1log⁡n(Δ−1)!

and

∑n=1N((log⁡n)α/(njnΔ−1))ERnI(1≤Rn≤cn)(log⁡N)α+2∼1(α+2)(Δ−1)!.

By checking directly the conditions in the proof of Theorem 2.2, the conclusion (3.5) can be obtained.

For the case j_n ∼ log n, let c_n = n(log n)^Δ+1, then it is not difficult to check that cn−1/jn→e−1 as n → ∞. Hence we get

ERnI(1≤Rn≤cn)∼jnΔ−1(Δ−1)!∫1cnx−1(1−x−1/jn)Δ−1dx=jnΔ−1(Δ−1)!∫1cnx−1∑k=0Δ−1Δ−1k(−1)kx−k/jndx=jnΔ−1(Δ−1)!log⁡cn+∑k=1Δ−1Δ−1k(−1)k(k/jn)−1(1−cn−k/jn)∼(log⁡n)Δ−1(Δ−1)!log⁡n+∑k=1Δ−1Δ−1k(−1)klog⁡nk(1−e−k)=(log⁡n)Δ(Δ−1)!1+∑k=1Δ−11kΔ−1k(−1)k(1−e−k)=γΔ(log⁡n)Δ(Δ−1)!

and

∑n=1N((log⁡n)α/(njnΔ−1))ERnI(1≤Rn≤cn)(log⁡N)α+2∼∑n=1N((log⁡n)α/(njnΔ−1))γΔ(log⁡n)Δ(log⁡N)α+2(Δ−1)!∼γΔ(α+2)(Δ−1)!.

By checking the conditions in the proof of Theorem 2.2 where j_n ∼ log n, the conclusion (3.6) can be obtained.□

Example 3.4

For p_nj_n = 1, Δ ≥ 2 and α > –2, let r_N = N and bN=1(log⁡N)δ for any δ > 0. If j_n = o(log n), then

∑N=1∞1(log⁡N)δP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−1(α+2)(Δ−1)!≥N<∞.

If j_n ∼ log n, then

∑N=1∞1(log⁡N)δP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−γΔ(α+2)(Δ−1)!≥N<∞

where

γΔ=1+∑k=1Δ−1Δ−1k(−1)k(1−e−k)k.

Example 3.5

For p_nj_n = 1, Δ ≥ 2 and α > –2, let rN=1loglog⁡N and bN=1N(loglog⁡N)δ+4 for any δ > 0. If j_n = o(log n), then

∑N=1∞1N(loglog⁡N)δ+4P∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−1(α+2)(Δ−1)!≥1loglog⁡N<∞.

If j_n ∼ log n, then

∑N=1∞1N(loglog⁡N)δ+4P∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−γΔ(α+2)(Δ−1)!≥1loglog⁡N<∞.

Example 3.6

Let p_nj_n = 1, Δ ≥ 2 and α > –2. Let r_N = r be a constant and bN=1N(log⁡N)δ, where 0 < δ≤ 1. If j_n = o(log n), then for r > 0,

∑N=1∞1N(log⁡N)δP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−1(α+2)(Δ−1)!≥r<∞.

If j_n ∼ log n, then

∑N=1∞1N(log⁡N)δP∑n=1N((log⁡n)α/(njnΔ−1))Rn(log⁡N)α+2−γΔ(α+2)(Δ−1)!≥r<∞.

Corollary 3.3

Let {b_N, N ≥ 1} and {r_N, N ≥ 1} satisfy the condition (2.5) and (2.6). If p_nj_n = 1, j_n ∼ (log n)^a, where 1 < a < 2, Δ = 2 and α > –3, then

∑N=1∞bNP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+3−12(α+3)≥rN<∞.

Proof

We can check directly the conditions in the proof of Theorem 2.3 to get the corollary. Since

ERnI(1≤Rn≤cn)∼jn∫1cnx−1(1−x−1/jn)dx =jn∫1cn(x−1−x−1−1/jn)dx =jn[log⁡cn+jn(cn−1/jn−1)] ∼(log⁡n)a(log⁡n+3loglog⁡n)+(log⁡n)2a[(n(log⁡n)3)−1/(log⁡n)a−1],

by Lemma 2.3 with t = log n, we have

ERnI(1≤Rn≤cn)∼ta(t+3log⁡t)+t2a[(ett3)−1/ta−1] =t2ta−1[1+3t−1log⁡t+ta−1((ett3)−1/ta−1)] ∼t2/2=(log⁡n)2/2.

Thus we have

∑n=1N((log⁡n)α/n)ERnI(1≤Rn≤cn)(log⁡N)α+3∼12(α+3). (3.7)

Then the conclusion can be obtained.□

Example 3.7

For p_nj_n = 1, j_n ∼ (log n)^a, where 1 < a < 2, Δ = 2 and α > –3, let r_N = N and bN=1(log⁡N)δ for any δ > a – 1. Then we have

∑N=1∞1(log⁡N)δP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+3−12(α+3)≥N<∞.

Example 3.8

For p_nj_n = 1, j_n ∼ (log n)^a, where 1 < a < 2, Δ = 2 and α > –3, let rN=1loglog⁡N and b_N = 1N(log⁡N)a−1(loglog⁡N)δ+4 for any δ > 0. Then we have

∑N=1∞1N(log⁡N)a−1(loglog⁡N)δ+4P∑n=1N((log⁡n)α/n)Rn(log⁡N)α+3−12(α+3)≥1loglog⁡N<∞.

Example 3.9

If p_nj_n = 1, j_n ∼ (log n)^a, where 1 < a < 2, Δ = 2 and α > –3, let r_N = r be a constant and bN=1N(log⁡N)δ, where a – 1 < δ ≤ 1. Then for r > 0, we have

∑N=1∞1N(log⁡N)δP∑n=1N((log⁡n)α/n)Rn(log⁡N)α+3−12(α+3)≥r<∞.

4 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 firstly get the complete convergence of the weighted sums with different forms by discussing three different 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.

Acknowledgement

The authors are very grateful to the referees for their valuable reports which improved the presentation of this work.

This work is supported by IRTSTHN (14IRTSTHN023), NSFC (11471104).

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Received: 2019-01-22

Accepted: 2019-03-14

Published Online: 2019-06-06

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0035

Keywords for this article

Ratios; Pareto distribution; complete convergence

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