## RESEARCH ARTICLE

# On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings

**Emad-Eldin A. A. Aly**

^{1, *}^{1}Department of Statistics & Operations Research, Faculty of Science, Kuwait University, Safat 13060, Kuwait

### Article Information

#### Identifiers and Pagination:

**Year:**2020

**Volume:**10

**First Page:**1

**Last Page:**7

**Publisher Id:**TOSPJ-10-1

**DOI:**10.2174/1876527002010010001

#### Article History:

**Received Date:**02/02/2019

**Revision Received Date:**06/11/2019

**Acceptance Date:**16/11/2019

**Electronic publication date:**14/02/2020

**Collection year:**2020

**© 2020 Emad-Eldin A. A. Aly.**

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

## Abstract

### Objectives:

To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution.

### Methods:

Earlier results on the distribution of the uniform incremental randomly weighted sums.

### Methods:

Based on theorems on weak and strong approximations of partial sum processes.

### Results and conculsions:

Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.

**Keywords:**Uniform spacings, Weak convergence, Gaussian process, Incremental asymptotic convergence, Random Sample, k spacings.

## 1. INTRODUCTION

Let
be the order statistics of a random sample of size (*n*-1) from the *U*(0,1) distribution. Let *k*=1,2, ... be arbitrary but fixed and assume that *n*=*mk*. The *U*(0,1) k-spacings are defined as

Let *X*_{1}, *X*_{2},... be *iidrv* with *E*(*X*_{i})=*µ*, Var(*X*_{i})=*ó*^{2}<∞ and common distribution function F(.). Assume that the Xi’s are independent of the Ui's. Define

where [s] is the integer part of s and *r*>0 is fixed.

Looking at *S _{m}* (

*t,k,r,F*) of (2) as a weighted partial sum of the

*X's*, Van Assche [1] obtained the exact distribution of

*S*

_{2}(1, 1,1,

*F*). Johnson and Kotz [2] studied some generalizations of Van Assche results. Soltani and Homei [3] considered the finite sample distribution of

*S*(

_{n}*1,1,1, F*). Soltani and Roozegar [4] considered the finite sample distribution of a case similar to

*S*(

_{m}*1,k,1, F*) in which the spacings (1) are not equally spaced. It is interesting to note that

*S*(

_{m}*t,k,r, F*) of (2) is also a randomly weighted partial sum of powers of k-spacings from the U(0,1) distribution.

Here, we will obtain the asymptotic distribution of the stochastic process

(3)where

(4)and Γ(.) is the gamma function.

The motivations and justifications of this work are given next. First, as noted by Johnson and Kotz [2], *S _{2}* (

*1,1,1, F*) is a random mixture of distributions and as such it has numerous applications in Sociology and in Biology. Second, the asymptotic theory of

*S*(

_{m}*t,k,r, F*) is a generalization of important results of Kimball [5], Darling [6], LeCam [7], Sethuraman and Rao [8], Koziol [9], Aly [10] and Aly [11] for sums of powers of spacings from the

*U*(0,1) distribution. Finally, we solve the open problem of proving the asymptotic normality of

*S*(

_{m}*1,k,1, F*) proposed by Soltani and Roozegar [4].

## 2. METHODS

### 2.1. The asymptotic distribution of αm (., *k,r, F*)

Let *Y*_{1},*Y*_{2},... be *iidrv* with the exponential distribution with mean 1 which are independent of the *Xi's*. By Proposition 13.15 of Breiman [12] we have for each *n*,

Hence, for each *m*,

where for 1≤*i*≤*m*,

are *iid Gamma* (*k*,1) random variables. Hence, for each *m*

and

(6)where

(7)Let *µ*_{l,k} be as in (4). Note that

and

The following Theorem will be needed in the sequel.

**Theorem A.** There exists a probability space on which a two-dimensional Wiener process
is defined such that

where *E W* (s)=0, *and*

Theorem A follows from the results of Einmahl [13], Zaitsev [14] and Götze and Zaitsev [15].

The main result of this paper is the following Theorem.

**Theorem 1. ***On some probability space, there exists a sequence of mean zero Gaussian processes Γ _{m}(t, k, r, F), 0≤t≤1 such that*

*where
for each m, and*

Theorem 1 follows from (6) and the following Theorem.

**Theorem 2. ***On the probability space of Theorem A,*

*where W (.) is as in (9)*.

**Proof of Theorem 2:** We will only prove here the case when *E*(*X*)=*µ*≠0. The case when *µ*=0 is straightforward and can be looked at as a special case of the case *µ*≠0. Note that

where

(15)It is clear that, uniformly in *t*, 0≤*t*≤1,

By (9), (15) and (16) we have, uniformly in *t*,0≤*t*≤1,

By Lemma 1.1.1 of Csörgö and Révész [17] we have, uniformly in *t*,0≤*t*≤1,

By (17) and (18) we have, uniformly in *t*,0≤*t*≤1,

By the LIL

(20)By (14), (19) and (20) we have, uniformly in *t*,0≤*t*≤1,

This proves (13).

**Corollary 1. *** By* (4), (8) *and* (12),

where

W(.) is a Wiener process, B(.) is a Brownian bridge and W(.) and B(.) are independent.

**Corollary 2. *** By* (11) *and* (21) *we have, as* m→∞,

and, in particular,

(23)Some special cases of (22) and (23) are given . For r=1 and k≥1,

and

and

where

## 3. RESULTS

In this section, we will use the same notation of Section 1

### 3.1. The scaled sum case

Define

and

We can prove that

where

(W_{1}(.),W_{2}(.),W_{3}(.))^{t} is a mean zero Gaussian vector with covariance (*t* Λ *s*) ∑ _{1} and

Let

We can show that

where *W*(.) is a Brownian Motion and *B*(.) is a Brownian bridge and *W*(.) and *B*(.) are independent. Consequently,

When *r*=1,*k*≥1

When *r*>0,*k*=1

### 3.2. The Centered Sum Process

Let and define

and

We can prove that

where

(*W*_{1}(.), *W*_{2}(.), *W*_{3}(.))^{t} is a mean zero Gaussian vector with covariance (*t* Λ *s*) ∑_{2} and

We can show that

where *W*(.) is a Brownian Motion and *B* (.) is a Brownian bridge and *W*(.) and *B*(.) are independent. Consequently,

When *r*=1,*k*≥1

When *r*> 0,*k*=1

### 3.3. The Renewal Process

For simplicity, we will consider the case of *r*=1. Define

and

By (5), for each *m*

Note that (see (3))

and hence, by Theorem 1

where *Γ*_{m} (., *k*, 1, *F*) is as in (11).

**Theorem 3. *** On the probability space of Theorem A,*

where

(25)*and W(.) is as in* (9).

Theorem 3 follows directly from (24) and the following Theorem.

**Theorem 4. *** On the probability space of Theorem A,*

*where Γ _{m}(t) is as in* (25).

**Proof:** By (7),

Note that

Hence

(26)where

and

By Theorem 2 and the LIL for Wiener processes,

(27)and

By a Lemma of Horváth [18]

and hence

(28)By the proof of Step 5 of Horváth [18] and Theorem 2 we can show that

(29)As to *E*_{m3},

where

and

By (28) and Lemma 1.1.1 of Csörgö and Révész [17] we have, uniformly in *t*,0≤*t*≤1,

By (28) and the LIL for Wiener processes,

(32)By (30)-(32),

(33)By (26)-(33) we obtain Theorem 4.

## 4. THE RANDOM VECTOR CASE

Let X_{1}, X_{2},... be *iid* random vectors with
and
Assume that the *Ui's* and the *R _{i,k}s* are same as in Section 1 and are independent of X

_{1}, X

_{2},... Define

and

Theorem 5 is a generalization of Theorem 1.

**Theorem 5.** On some probability space, there exists a mean zero sequence of Gaussian processes
such that

where, for each m,

and

**Corollary 1** *. *By* (11) *and* (21) *we have, as* m→∞,

and, in particular,

where

Particular cases of Corollary 1* are given next.

For *r* = 1 and *k* ≥ 1,

and

For *r* > 0 and *k* = 1,

and

where

## CONCLUSION

We proved the weak convergence of a stochastic process defined in terms of partial sums of randomly weighted powers of uniform spacings. The asymptotic results of several important generalizations and special cases are given.

### CONSENT FOR PUBLICATION

Not applicable.

### AVAILABILITY OF DATA AND MATERIALS

Not applicable.

### FUNDING

None.

### CONFLICT OF INTEREST

The author declare no conflict of interest, financial or otherwise.

### ACKNOWLEDGEMENTS

Declared none.