On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings
Emad-Eldin A. A. Aly1, *
Identifiers and Pagination:Year: 2020
First Page: 1
Last Page: 7
Publisher Id: TOSPJ-10-1
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
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.
To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution.
Earlier results on the distribution of the uniform incremental randomly weighted sums.
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.
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 X1, X2,... be iidrv with E(Xi)=µ, Var(Xi)=ó2<∞ 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 Sm (t,k,r,F) of (2) as a weighted partial sum of the X's, Van Assche  obtained the exact distribution of S2 (1, 1,1, F). Johnson and Kotz  studied some generalizations of Van Assche results. Soltani and Homei  considered the finite sample distribution of Sn (1,1,1, F). Soltani and Roozegar  considered the finite sample distribution of a case similar to Sm (1,k,1, F) in which the spacings (1) are not equally spaced. It is interesting to note that Sm (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
and Γ(.) is the gamma function.
The motivations and justifications of this work are given next. First, as noted by Johnson and Kotz , S2 (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 Sm (t,k,r, F) is a generalization of important results of Kimball , Darling , LeCam , Sethuraman and Rao , Koziol , Aly  and Aly  for sums of powers of spacings from the U(0,1) distribution. Finally, we solve the open problem of proving the asymptotic normality of Sm (1,k,1, F) proposed by Soltani and Roozegar .
2.1. The asymptotic distribution of αm (., k,r, F)
Let Y1,Y2,... be iidrv with the exponential distribution with mean 1 which are independent of the Xi's. By Proposition 13.15 of Breiman  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
Let µl,k be as in (4). Note that
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
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 Γm(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
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  we have, uniformly in t,0≤t≤1,
By (17) and (18) we have, uniformly in t,0≤t≤1,
By the LIL
By (14), (19) and (20) we have, uniformly in t,0≤t≤1,
This proves (13).
Corollary 1. By (4), (8) and (12),
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,
Some special cases of (22) and (23) are given . For r=1 and k≥1,
In this section, we will use the same notation of Section 1
3.1. The scaled sum case
We can prove that
(W1(.),W2(.),W3(.))t is a mean zero Gaussian vector with covariance (t Λ s) ∑ 1 and
We can show that
where W(.) is a Brownian Motion and B(.) is a Brownian bridge and W(.) and B(.) are independent. Consequently,
3.2. The Centered Sum Process
Let and define
We can prove that
(W1(.), W2(.), W3(.))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> 0,k=1
3.3. The Renewal Process
For simplicity, we will consider the case of r=1. Define
By (5), for each m
Note that (see (3))
and hence, by Theorem 1
where Γm (., k, 1, F) is as in (11).
Theorem 3. On the probability space of Theorem A,
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 Γm(t) is as in (25).
Proof: By (7),
By Theorem 2 and the LIL for Wiener processes,
By a Lemma of Horváth 
By the proof of Step 5 of Horváth  and Theorem 2 we can show that
As to Em3,
By (28) and Lemma 1.1.1 of Csörgö and Révész  we have, uniformly in t,0≤t≤1,
By (28) and the LIL for Wiener processes,
By (26)-(33) we obtain Theorem 4.
4. THE RANDOM VECTOR CASE
Let X1, X2,... be iid random vectors with and Assume that the Ui's and the Ri,ks are same as in Section 1 and are independent of X1, X2,... Define
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,
Corollary 1 *. By (11) and (21) we have, as m→∞,
and, in particular,
Particular cases of Corollary 1* are given next.
For r = 1 and k ≥ 1,
For r > 0 and k = 1,
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.
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