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.
Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.
Open Peer Review Details | |||
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Manuscript submitted on 02-02-2019 |
Original Manuscript | On The Distribution of Partial Sums of Randomly Weighted 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
(1)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
(2)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 [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 S_{2} (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 S_{n} (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 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 [2N.L. Johnson, and S. Kotz, "Randomly weighted averages: Some aspects and extensions", Am. Stat., vol. 44, no. 3, pp. 245-249.], 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 [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 S_{m} (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] ].
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 [12L. Breiman, Probability., Addison-Wesley: Reading, Massachusetts, .] 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
(5)and
(6)where
(7)Let µ_{l,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 Γ_{m}(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
In this section, we will use the same notation of Section 1
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
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
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 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 [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
As to E_{m3},
(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.
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
Not applicable.
Not applicable.
None.
The author declare no conflict of interest, financial or otherwise.
Declared none.
[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] |