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

where NN is a standard normal variable and μμ = μμnn and γγ = ττnn/μμnn are the common mean and the coefficient of variation of the r.v.’s. e convergence in (2) was generalized and extended bymany authors in different ways. For example, Zhang and Huang [3] studied the functional version of this result. ey proved under general conditions, not involving the dependence structure of the sequence, that


Introduction
Let (  )  be a sequence of independent, positive, and square-integrable random variables (r.v.'s) de�ned on some probability space (Ω  ).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.e convergence in (2) was generalized and extended by many authors in different ways.For example, Zhang and Huang [3] here and in the sequel () denotes the Wiener process.e functional version of the convergence of products of sums was also studied by Kosi ń ski [4], for i.i.d.sequences belonging to the domain of attraction of the -stable law with   ( 2.e �rst result for sequences of nonidentically distributed r.v.'s was obtained by Matuła and Stȩpie ń [5].Similarly as in the invariance principle, the study of the non-i. and   () =  for  such that   () = , where   = ( 2  / 2  )( 2  / 2  )  .In [5] it was proved that for a sequence of independent, positive, and square-integrable random variables satisfying the Lindeberg condition and such that ∑ ∞   2  /(  ) 2 < ∞, the process   () converges weakly in the space   to the process exp(∫   ()()).is 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 arti�cial de�nition in the case   ()   and replace the strange shied index in  2   .In this setting, the imposed conditions and obtained results will be much more natural.

Main Results
Let ℋ be the family of functions  3 but   may be considered as a sum of i.i.d.(Poisson) r.v.'s.us, from the well-known moment bounds, there exists a constant    not depending on  such that   −    3 ≤    3/2 , for every   .erefore, the Lyapunov condition may be easily veri�ed.e condition (10) is also satis�ed.We apply our eorem 1 with ()  /, then    (

Proofs and Auxiliary Results
In the proof of our main result, we shall use the following lemma.
Lemma Proof.e proof is based on the inequality   ≤   , which may be found on page 150 in [7].e inequality (b) is proved, for example, in eorem 11.3.5 on page 397 in [8].
Proof of eorem 1.In the �rst step we shall prove that which are continuously differentiable on (  that is     ( , furthermore, such that Example 2. Let (  )  be a sequence of independent r.v.'s with Poisson distribution Po(  ) with parameter    .Let us take       , then   ≥  almost surely,       and  2 Let us observe that   −    3    − ()  ()  in     as  ⟶ ∞.(11)In order to illustrate our result let us present two examples.  Var(  )  .Moreover,    2⋯()  ( 3)/2 and  2   Var      ⋯    (  )/2.
, where   .In this case      and we get 2  / 2  )   (  3)/(  ) and we have the following weak convergence in the space  , as provided   −3/2.erefore, we have the following example.Example 3. Let (  )  be a sequence of i.i.d.r.v.'s with the standard exponential distribution.en    Var(  )   and  2   .We apply eorem 1 with    and the function 4.Let   be the metric making the space   to be a complete and separable metric space (Polish space) and let      p ≤≤ || be the uniform metric.
Furthermore, denote by  LP the Lévy-Prohorov metric in the space of probability measures on   and by  KF the Ky-Fan metric.Let , , and   be random elements with values in  , then (a) if     ≤ , then  KF   ≤ √, (b)  LP   ≤  KF  , (c) if        almost surely, then        .