Necessary and sufficient conditions for weak convergence of random sums of independent random variables

Andrzej Krajka; Zdzisław Rychlik

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 3, page 465-482
  • ISSN: 0010-2628

Abstract

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Let { X n , n ≥ 1 } be a sequence of independent random variables such that E X n = a n , E ( X n - a n ) 2 = σ n 2 , n ≥ 1 . Let { N n , n ≥ 1 } be a sequence od positive integer-valued random variables. Let us put S N n = ∑ k = 1 N n X k , L n = ∑ k = 1 n a k , s n 2 = ∑ k = 1 n σ k 2 , n ≥ 1 . In this paper we present necessary and sufficient conditions for weak convergence of the sequence { ( S N n - L n ) / s n , n ≥ 1 } , as n → ∞ . The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

How to cite

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Krajka, Andrzej, and Rychlik, Zdzisław. "Necessary and sufficient conditions for weak convergence of random sums of independent random variables." Commentationes Mathematicae Universitatis Carolinae 34.3 (1993): 465-482. <http://eudml.org/doc/247521>.

@article{Krajka1993,
abstract = {Let $\lbrace X_n,\, n\ge 1\rbrace $ be a sequence of independent random variables such that $EX_n=a_n$, $E(X_n-a_n)^2=\sigma _n^2$, $n\ge 1$. Let $\lbrace N_n,\, n\ge 1\rbrace $ be a sequence od positive integer-valued random variables. Let us put $S_\{N_n\}=\sum _\{k=1\}^\{N_n\} X_k$, $L_n=\sum _\{k=1\}^\{n\} a_k$, $s_n^2=\sum _\{k=1\}^\{n\} \sigma _k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\lbrace (S_\{N_n\}-L_n)/s_n,\, n\ge 1\rbrace $, as $n\rightarrow \infty $. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).},
author = {Krajka, Andrzej, Rychlik, Zdzisław},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {random sums; weak convergence; stable law; nonrandom centering; measure of dependence between $\sigma $-fields; random sums; stable law; nonrandom centering; measure of dependence; weak convergence},
language = {eng},
number = {3},
pages = {465-482},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Necessary and sufficient conditions for weak convergence of random sums of independent random variables},
url = {http://eudml.org/doc/247521},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Krajka, Andrzej
AU - Rychlik, Zdzisław
TI - Necessary and sufficient conditions for weak convergence of random sums of independent random variables
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 465
EP - 482
AB - Let $\lbrace X_n,\, n\ge 1\rbrace $ be a sequence of independent random variables such that $EX_n=a_n$, $E(X_n-a_n)^2=\sigma _n^2$, $n\ge 1$. Let $\lbrace N_n,\, n\ge 1\rbrace $ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}=\sum _{k=1}^{N_n} X_k$, $L_n=\sum _{k=1}^{n} a_k$, $s_n^2=\sum _{k=1}^{n} \sigma _k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\lbrace (S_{N_n}-L_n)/s_n,\, n\ge 1\rbrace $, as $n\rightarrow \infty $. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).
LA - eng
KW - random sums; weak convergence; stable law; nonrandom centering; measure of dependence between $\sigma $-fields; random sums; stable law; nonrandom centering; measure of dependence; weak convergence
UR - http://eudml.org/doc/247521
ER -

References

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  2. Bradley C.R., Bryc W., Janson S., Remarks on the foundations of measures of dependence, New Perspectives in Theoretical and Applied Statistics, ed. by Dr. Madan L. Puri, Dr. Jose Perez Vilaplana and Dr. Wolfgang Wertz, John Wiley & Sons Inc., 1987, pp. 421-437. Zbl0619.60011MR0900202
  3. Finkelstein M., Tucker H.G., A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering, Proc. Amer. Math. Soc. 107 (1989), 1061-1070. (1989) Zbl0682.60017MR0993749
  4. Krajka A., Rychlik Z., On the limit behaviour of randomly indexed sums of independent random variables, Liet. Matem. Rink. 28 (1988), 484-494. (1988) Zbl0659.60040MR0969463
  5. Krajka A., Rychlik Z., On the rate of convergence in the random central limit theorem in Hilbert space, Probab. and Math. Stat. 11 (1990), 97-108. (1990) Zbl0741.60006MR1096941
  6. Kruglov V.M., O skhodimosti raspredelenii summ sluchainogo chisla nezavisimykch sluchainykh velichin k normal'nomu raspredeleniyu, Vestnik Mosk. Univ. 5 (1976), 5-12. (1976) MR0426104
  7. Petrov V.V., Predel'nye teoremy dlja summ nezavisimykh sluchainykh velichin, Moskva, Nauka, 1987. MR0896036
  8. Rychlik Z., A remainder term estimate in a random-sum central limit theorem, Bull. of the Pol. Acad. of Sci., Math. XXV (1985), 57-63. (1985) Zbl0564.60024MR0798728
  9. Szasz D., Freyer B., On the sums of a random number of random variables, Liet. Matem. Rink. 11 (1971), 181-187. (1971) Zbl0229.60035MR0303582

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