| Title:
|
Necessary and sufficient conditions for weak convergence of random sums of independent random variables (English) |
| Author:
|
Krajka, A. |
| Author:
|
Rychlik, Z. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
465-482 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\{X_n,\, n\geq 1\}$ be a sequence of independent random variables such that $EX_n=a_n$, $E(X_n-a_n)^2=\sigma _n^2$, $n\geq 1$. Let $\{N_n,\, n\geq 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}=\sum_{k=1}^{N_n} X_k$, $L_n=\sum_{k=1}^{n} a_k$, $s_n^2=\sum_{k=1}^{n} \sigma _k^2$, $n\geq 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,\, n\geq 1\}$, as $n\rightarrow \infty $. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989). (English) |
| Keyword:
|
random sums |
| Keyword:
|
weak convergence |
| Keyword:
|
stable law |
| Keyword:
|
nonrandom centering |
| Keyword:
|
measure of dependence between $\sigma $-fields |
| MSC:
|
60F05 |
| MSC:
|
60G50 |
| idZBL:
|
Zbl 0785.60016 |
| idMR:
|
MR1243079 |
| . |
| Date available:
|
2009-01-08T18:05:25Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118604 |
| . |
| Reference:
|
[1] Bhattacharya R.N., Ranga Rao R.: Normal approximation and Asymptotic Expansions.John Wiley & Sons, New York-London-Sydney-Toronto, 1976. Zbl 0657.41001, MR 0436272 |
| Reference:
|
[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. Zbl 0619.60011, MR 0900202 |
| Reference:
|
[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. Zbl 0682.60017, MR 0993749 |
| Reference:
|
[4] Krajka A., Rychlik Z.: On the limit behaviour of randomly indexed sums of independent random variables.Liet. Matem. Rink. 28 (1988), 484-494. Zbl 0659.60040, MR 0969463 |
| Reference:
|
[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. Zbl 0741.60006, MR 1096941 |
| Reference:
|
[6] Kruglov V.M.: O skhodimosti raspredelenii summ sluchainogo chisla nezavisimykch sluchainykh velichin k normal'nomu raspredeleniyu.Vestnik Mosk. Univ. 5 (1976), 5-12. MR 0426104 |
| Reference:
|
[7] Petrov V.V.: Predel'nye teoremy dlja summ nezavisimykh sluchainykh velichin.Moskva, Nauka, 1987. MR 0896036 |
| Reference:
|
[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. Zbl 0564.60024, MR 0798728 |
| Reference:
|
[9] Szasz D., Freyer B.: On the sums of a random number of random variables.Liet. Matem. Rink. 11 (1971), 181-187. Zbl 0229.60035, MR 0303582 |
| . |
