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Journal of Mathematics
Volume 2013 (2013), Article ID 640137, 6 pages
Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables
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.
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.
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  obtained the central limit theorem for sums of records. Rempała and Wesołowski  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  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 , 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 . 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  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  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 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
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 ) 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 ). 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 .
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.
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