Journal of Mathematics

Volume 2013 (2013), Article ID 640137, 6 pages

http://dx.doi.org/10.1155/2013/640137

Research Article

Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables

Przemysław Matuła and Iwona Stępień

Institute of Mathematics, Maria Curie-Skłodowska University, Plac M.C.-Skłodowskiej 1, 20-031 Lublin, Poland

Received 27 August 2012; Accepted 11 November 2012

Academic Editor: Inés Couso

Copyright © 2013 Przemysław Matuła and Iwona Stępień. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the weak convergence in the space of processes constructed from products of sums of independent but not necessarily identically distributed random variables. The presented results extend and generalize limit theorems known so far for i.i.d. sequences.

1. Introduction

Let be a sequence of independent, positive, and square-integrable random variables (r.v.’s) defined on some probability space . For every let us put

Arnold and Villaseñor [1] obtained the central limit theorem for sums of records. Rempała and Wesołowski [2] observed that this result was of general nature and proved that for a sequence of i.i.d. positive and square-integrable r.v.’s one has

where is a standard normal variable and and are the common mean and the coefficient of variation of the r.v.’s. The convergence in (2) was generalized and extended by many authors in different ways. For example, Zhang and Huang [3] studied the functional version of this result. They proved under general conditions, not involving the dependence structure of the sequence, that

here and in the sequel denotes the Wiener process. The functional version of the convergence of products of sums was also studied by Kosiński [4], for i.i.d. sequences belonging to the domain of attraction of the -stable law with . The first result for sequences of nonidentically distributed r.v.’s was obtained by Matuła and Stȩpień [5]. Similarly as in the invariance principle, the study of the non-i.i.d. sequences requires defining the processes in a different way than in (3). For and each let us define the function , where . Let us introduce the process

and for such that , where . In [5] it was proved that for a sequence of independent, positive, and square-integrable random variables satisfying the Lindeberg condition and such that , the process converges weakly in the space to the process . This result was proved in one theorem for the function and in the second for functions .

Our goal is to prove the aforementioned result for a large class of continuous functions on , which may be unbounded. Furthermore, we shall slightly change the process in order to avoid its artificial definition in the case and replace the strange shifted index in . In this setting, the imposed conditions and obtained results will be much more natural.

2. Main Results

Let be the family of functions which are continuously differentiable on that is,, furthermore, such that

Let us observe that from (5) it follows that is integrable, if this function is also monotonic then, in consequence, (6) is satisfied. It is obvious that . The family also contains functions of the form with and with and the value at 0 is arbitrary.

For any let us define the process

where and we also take . In this way we obtain a sequence of processes in such that

recall that we have defined . Under our assumptions the trajectories of the process are almost surely positive functions.

Our main result concerning weak convergence of in the space is as follows.

Theorem 1. Let be a sequence of independent, positive, and square-integrable r.v.’s satisfying the Lindeberg condition:

and such that

Then for any

In order to illustrate our result let us present two examples.

Example 2. Let be a sequence of independent r.v.’s with Poisson distribution with parameter . Let us take , then almost surely, and . Moreover, and =. Let us observe that but may be considered as a sum of i.i.d. (Poisson) r.v.’s. Thus, from the well-known moment bounds, there exists a constant not depending on such that , for every . Therefore, the Lyapunov condition may be easily verified. The condition (10) is also satisfied. We apply our Theorem 1 with , then and we have the following weak convergence in the space , as

It may be easily checked (see page  42 in [6] for details) that the integral has Gaussian distribution with

provided . Therefore, we have the following example.

Example 3. Let be a sequence of i.i.d. r.v.’s with the standard exponential distribution. Then and . We apply Theorem 1 with and the function , where . In this case and we get

3. Proofs and Auxiliary Results

In the proof of our main result, we shall use the following lemma.

Lemma 4. Let be the metric making the space to be a complete and separable metric space (Polish space) and let be the uniform metric. Furthermore, denote by the Lévy-Prohorov metric in the space of probability measures on and by the Ky-Fan metric. Let , , and be random elements with values in , then

(a)if , then ,(b), (c)if almost surely, then .

Proof. The proof is based on the inequality , which may be found on page  150 in [7]. The inequality (b) is proved, for example, in Theorem 11.3.5 on page  397 in [8].

Proof of Theorem 1. In the first step we shall prove that

where in , as . We shall apply the expansion of the logarithm , for , where . Let us put , then we easily get

Let us observe that

From the assumption (10) and the Kronecker lemma it follows that . It is easy to see that from the Lindeberg condition (9), the Feller condition

holds; therefore,

Moreover, by our assumption (6) we get

By the above remarks and Toeplitz theorem on the transformation of sequences into sequences (Problem 2.3.1 in [9]) we obtain

Thus from (17) we have in , as .

From (10) the strong law of large numbers holds, that is, almost surely. Thus, for almost all and sufficiently large there holds . Since that

Consequently in , and (15) is proved.

It remains to prove that

in , as .

Let be given and let us define a process

Let us observe that

Since from the definition of we have , therefore, by the integrability of the function and Lemma 4(a) we get

Let us define a continuous mapping

The above integral is properly defined since is bounded and . In the next step we shall prove that

where in , as and

At first let us observe that

by the Feller condition (18).

In the case , let us calculate the integral

In fact does not depend on , thus

as , since and .

Let us find the bounds for the second term :

Further, for any , we have

by the Kolmogorov inequality and the Feller condition (18).

Finally, let be sufficiently large, that is, such that , then

where . Thus, again by the Kolmogorov inequality and the Feller condition, we get

Consequently, from (30), (32), (34), and (36) the statement (28) follows.

To end the proof, let us note that for the process defined by (29), we have in , by the result of Prohorov (see Problem 1, page  77 in [7]). In fact, Prohorov considered a broken-line process and its convergence in , but

from the Lindeberg condition (9). Thus, by the continuity of , we get

Furthermore,

by (5) and the properties of the modulus of continuity of the Wiener process, that is,

see formula (7) on page  534 in [6].

For short, let us put . Then, with the notation introduced in Lemma 4, we get

Now, (23) follows from (26), (28), (38), and (39) and the proof is completed.

References

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