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Contents of PMS, Vol. 30, Fasc. 1,
pages 73 - 85
 

LIMIT THEOREMS FOR PRODUCTS OF SUMS OF INDEPENDENT RANDOM VARIABLES

Tomasz K. Krajka
Zdzisław Rychlik

Abstract: Let (X ,n > 1)
  n be a sequence of independent random variables with finite second moments and (N , n > 1)
  n be a sequence of positive integer-valued random variables. Write S  =  sum n  (X  -EX  ),n > 1,
 n     k=1  k    k and let N be a standard normal random variable. In the paper the convergences

  n                          Nn
( prod  (S /a + 1))gn-D-->  eN and ( prod  (S /a + 1))gn-D-->  eN
 k=1  k  k                   k=1  k  k
are considered for some sequences (an) and (gn) of positive integer numbers such that Sn + an > 0 a.e. The case when gn  are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3].

2000 AMS Mathematics Subject Classification: Primary: 60F05; Secondary: 60G50.

Keywords and phrases: Lognormal distribution; randomly indexed product of sums; central limit theorem; self-normalized.

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