# Article

Full entry | PDF   (0.2 MB)
Keywords:
random sums; weak convergence; stable law; nonrandom centering; measure of dependence between $\sigma$-fields
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).
References:
[1] Bhattacharya R.N., Ranga Rao R.: Normal approximation and Asymptotic Expansions. John Wiley & Sons, New York-London-Sydney-Toronto, 1976. MR 0436272 | Zbl 0657.41001
[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. MR 0900202 | Zbl 0619.60011
[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. MR 0993749 | Zbl 0682.60017
[4] Krajka A., Rychlik Z.: On the limit behaviour of randomly indexed sums of independent random variables. Liet. Matem. Rink. 28 (1988), 484-494. MR 0969463 | Zbl 0659.60040
[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. MR 1096941 | Zbl 0741.60006
[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
[7] Petrov V.V.: Predel'nye teoremy dlja summ nezavisimykh sluchainykh velichin. Moskva, Nauka, 1987. MR 0896036
[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. MR 0798728 | Zbl 0564.60024
[9] Szasz D., Freyer B.: On the sums of a random number of random variables. Liet. Matem. Rink. 11 (1971), 181-187. MR 0303582 | Zbl 0229.60035

Partner of