ISSN: 2666-1489 ― Volume 10, 2020

RESEARCH ARTICLE

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-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.

Tel: +965 66655671; E-mail: eealy50@gmail.com

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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

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}* (

[http://dx.doi.org/10.1016/j.spl.2009.01.009] ] considered the finite sample distribution of

[http://dx.doi.org/10.1016/j.spl.2012.02.007] ] considered the finite sample distribution of a case similar to

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}* (

[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",

[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",

[http://dx.doi.org/10.1214/aos/1176343058] ], Koziol [9J.A. Koziol, "A note on limiting distributions for spacings statistics",

[http://dx.doi.org/10.1007/BF00533817] ], Aly [10E-E.A.A. Aly, "Some limit theorems for uniform and exponential spacings",

[http://dx.doi.org/10.2307/3314641] ] for sums of powers of spacings from the

[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*

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 [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*

*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 [17M. Csörgö, and P. Revesz, *P, Strong Approximations in Probability and Statistics.**, *Academic Press: New York, .] 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

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*

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},

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,

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

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.

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[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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