The Open Mathematics, Statistics and Probability Journal


The Open Statistics & Probability Journal

ISSN: 2666-1489 ― Volume 10, 2020
RESEARCH ARTICLE

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



Emad-Eldin A. A. Aly1, *
1 Department of Statistics & Operations Research, Faculty of Science, Kuwait University, Safat 13060, Kuwait

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.


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.


* Address correspondence to this author at Department of Statistics & Operations Research Faculty of Science, Kuwait University, Safat 13060, Kuwait;
Tel: +965 66655671; E-mail: eealy50@gmail.com






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

(1)

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

(2)

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 [1W. Van Assche, "A random variable uniformly distributed between two independent random variables", Sankhya A, vol. 49, no. 2, pp. 207-211.] obtained the exact distribution of S2 (1, 1,1, F). Johnson and Kotz [2N.L. Johnson, and S. Kotz, "Randomly weighted averages: Some aspects and extensions", Am. Stat., vol. 44, no. 3, pp. 245-249.] studied some generalizations of Van Assche results. Soltani and Homei [3A.R. Soltani, and H. Homei, "Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex", Stat. Probab. Lett., vol. 79, no. 9, pp. 1215-1218.
[http://dx.doi.org/10.1016/j.spl.2009.01.009]
] considered the finite sample distribution of Sn (1,1,1, F). Soltani and Roozegar [4A.R. Soltani, and R. Roozegar, "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach", Stat. Probab. Lett., vol. 82, no. 5, pp. 1012-1020.
[http://dx.doi.org/10.1016/j.spl.2012.02.007]
] 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

(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 [2N.L. Johnson, and S. Kotz, "Randomly weighted averages: Some aspects and extensions", Am. Stat., vol. 44, no. 3, pp. 245-249.], 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 [5H.E. Kimball, "On the asymptotic distribution of the sum of powers of unit frequency differences", Ann. Math. Stat., vol. 21, no. 2, pp. 263-271.
[http://dx.doi.org/10.1214/aoms/1177729843]
], Darling [6D.A. Darling, "On a class of problems related to the random division of an interval", Ann. Math. Stat., vol. 24, no. 2, pp. 239-253.
[http://dx.doi.org/10.1214/aoms/1177729030]
], LeCam [7L. LeCam, "Un theoreme sur la division d’un intervalle par des points pris au hasard", Publ. Inst. Stat. Univ. Paris, vol. 7, no. 3/4, pp. 7-16.], Sethuraman and Rao [8J. Sethuraman, and J.S. Rao, "Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors", Ann. Stat., vol. 3, no. 2, pp. 299-313.
[http://dx.doi.org/10.1214/aos/1176343058]
], Koziol [9J.A. Koziol, "A note on limiting distributions for spacings statistics", Z. Wahrsch. Verw. Gebiete, vol. 51, no. 1, pp. 55-62.
[http://dx.doi.org/10.1007/BF00533817]
], Aly [10E-E.A.A. Aly, "Some limit theorems for uniform and exponential spacings", Canad. J. Statist., vol. 11, no. 1, pp. 211-219.] and Aly [11E-E.A.A. Aly, "Strong approximations of quadratic sums of uniform spacings", Can. J. Stat., vol. 16, no. 2, pp. 201-207.
[http://dx.doi.org/10.2307/3314641]
] 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 [4A.R. Soltani, and R. Roozegar, "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach", Stat. Probab. Lett., vol. 82, no. 5, pp. 1012-1020.
[http://dx.doi.org/10.1016/j.spl.2012.02.007]
].

2. METHODS

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 [12L. Breiman, Probability., Addison-Wesley: Reading, Massachusetts, .] we have for each n,

Hence, for each m,

where for 1≤im,

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

(5)

and

(6)

where

(7)

Let µl,k be as in (4). Note that

(8)

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

(9)

where E W (s)=0, and

(10)

Theorem A follows from the results of Einmahl [13U. Einmahl, "Extension of results of Komlós, Major and Tusnády to the multivariate case", J. Mult. Anal., vol. 28, no. 1, pp. 20-68.
[http://dx.doi.org/10.1016/0047-259X(89)90097-3]
], Zaitsev [14A.Yu. Zaitsev, "Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments", ESAIM Probab. Stat., vol. 2, pp. 41-108.
[http://dx.doi.org/10.1051/ps:1998103]
] and Götze and Zaitsev [15F. Götze, and A.Yu. Zaitsev, "Bounds for the rate of strong approximation in the multidimensional invariance principle", Theory Probab. Appl., vol. 53, no. 1, pp. 100-123.].

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

(11)

where for each m, and

(12)

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

Theorem 2. On the probability space of Theorem A,

(13)

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

(14)

where

(15)

It is clear that, uniformly in t, 0≤t≤1,

(16)

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

(17)

By Lemma 1.1.1 of Csörgö and Révész [17M. Csörgö, and P. Revesz, P, Strong Approximations in Probability and Statistics., Academic Press: New York, .] we have, uniformly in t,0≤t≤1,

(18)

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

(19)

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),

(21)

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→∞,

(22)

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

(W1(.),W2(.),W3(.))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

(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=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

(24)

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,

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 Γm(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 [18L. Horváth, "Strong approximation of renewal processes", Stochastic Process. Appl., vol. 18, no. 1, pp. 127-138.
[http://dx.doi.org/10.1016/0304-4149(84)90166-2]
]

and hence

(28)

By the proof of Step 5 of Horváth [18L. Horváth, "Strong approximation of renewal processes", Stochastic Process. Appl., vol. 18, no. 1, pp. 127-138.
[http://dx.doi.org/10.1016/0304-4149(84)90166-2]
] and Theorem 2 we can show that

(29)

As to Em3,

(30)

where

and

By (28) and Lemma 1.1.1 of Csörgö and Révész [17M. Csörgö, and P. Revesz, P, Strong Approximations in Probability and Statistics., Academic Press: New York, .] we have, uniformly in t,0≤t≤1,

(31)

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 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

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.

REFERENCES

[1] W. Van Assche, "A random variable uniformly distributed between two independent random variables", Sankhya A, vol. 49, no. 2, pp. 207-211.
[2] N.L. Johnson, and S. Kotz, "Randomly weighted averages: Some aspects and extensions", Am. Stat., vol. 44, no. 3, pp. 245-249.
[3] A.R. Soltani, and H. Homei, "Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex", Stat. Probab. Lett., vol. 79, no. 9, pp. 1215-1218.
[http://dx.doi.org/10.1016/j.spl.2009.01.009]
[4] A.R. Soltani, and R. Roozegar, "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach", Stat. Probab. Lett., vol. 82, no. 5, pp. 1012-1020.
[http://dx.doi.org/10.1016/j.spl.2012.02.007]
[5] H.E. Kimball, "On the asymptotic distribution of the sum of powers of unit frequency differences", Ann. Math. Stat., vol. 21, no. 2, pp. 263-271.
[http://dx.doi.org/10.1214/aoms/1177729843]
[6] D.A. Darling, "On a class of problems related to the random division of an interval", Ann. Math. Stat., vol. 24, no. 2, pp. 239-253.
[http://dx.doi.org/10.1214/aoms/1177729030]
[7] L. LeCam, "Un theoreme sur la division d’un intervalle par des points pris au hasard", Publ. Inst. Stat. Univ. Paris, vol. 7, no. 3/4, pp. 7-16.
[8] J. Sethuraman, and J.S. Rao, "Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors", Ann. Stat., vol. 3, no. 2, pp. 299-313.
[http://dx.doi.org/10.1214/aos/1176343058]
[9] J.A. Koziol, "A note on limiting distributions for spacings statistics", Z. Wahrsch. Verw. Gebiete, vol. 51, no. 1, pp. 55-62.
[http://dx.doi.org/10.1007/BF00533817]
[10] E-E.A.A. Aly, "Some limit theorems for uniform and exponential spacings", Canad. J. Statist., vol. 11, no. 1, pp. 211-219.
[11] E-E.A.A. Aly, "Strong approximations of quadratic sums of uniform spacings", Can. J. Stat., vol. 16, no. 2, pp. 201-207.
[http://dx.doi.org/10.2307/3314641]
[12] L. Breiman, Probability., Addison-Wesley: Reading, Massachusetts, .
[13] U. Einmahl, "Extension of results of Komlós, Major and Tusnády to the multivariate case", J. Mult. Anal., vol. 28, no. 1, pp. 20-68.
[http://dx.doi.org/10.1016/0047-259X(89)90097-3]
[14] A.Yu. Zaitsev, "Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments", ESAIM Probab. Stat., vol. 2, pp. 41-108.
[http://dx.doi.org/10.1051/ps:1998103]
[15] F. Götze, and A.Yu. Zaitsev, "Bounds for the rate of strong approximation in the multidimensional invariance principle", Theory Probab. Appl., vol. 53, no. 1, pp. 100-123.
[16] M. Csörgö, and L. Horváth, Weighted Approximations in Probability and Statistics., John Wiley and Sons: New York, .
[17] M. Csörgö, and P. Revesz, P, Strong Approximations in Probability and Statistics., Academic Press: New York, .
[18] L. Horváth, "Strong approximation of renewal processes", Stochastic Process. Appl., vol. 18, no. 1, pp. 127-138.
[http://dx.doi.org/10.1016/0304-4149(84)90166-2]
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