# ON CONVERGENCE OF SERIES OF INDEPENDENTS RANDOM VARIABLES

• Sung, Soo-Hak (Department of Applied Mathematics, Paichai University) ;
• Volodin, Andrei-I. (Department of Mathematics and Statistics, University of Regina)
• Published : 2001.01.01
• 76 5

#### Abstract

The rate of convergence for an almost surely convergent series $S_n={\Sigma^n}_{i-1}X_i$ of independent random variables is studied in this paper. More specifically, when S$_{n}$ converges almost surely to a random variable S, the tail series $T_n{\equiv}$ S - S_{n-1} = {\Sigma^\infty}_{i-n} X_i$is a well-defined sequence of random variables with T$_{n}\rightarrow$0 almost surely. Conditions are provided so that for a given positive sequence {$b_n, n {\geq$1}, the limit law sup$_{\kappa}\geqn | T_{\kappa}|/b_n \rightarrow\$ 0 holds. This result generalizes a result of Nam and Rosalsky [4].

#### References

1. Teor. Imovirnost. ta Mat. Statist.;Ukrainian, English translation in Theor. Probab. Math. Statist v.52 On the rate of convergence of series of random variables E. Nam;A. Rosalsky
2. Stochastic Anal. Appl. v.16 On convergence of series of random variables with applications to martingale convergence and to convergence of series with orthogonal summands A. Rosalsky;J. Rosenblatt
3. On the convergence rate of series of indpendent random variables Researcj Developments in Probability and Statistics: Festschrift in Honor of Madan L. Puri on the Occasion of his 65th Birthday E. Nam;A. Rosalsky;E. Brunner(ed.);M. Denker(ed.)
4. Probability Theory: Independence, interchangeability, Martingales(2nd ed.) Y. S. Chow;H. Teicher
5. Sankhya Ser. A v.47 On some classical results probability theory N. Etemadi
6. Stochastic Anal. Appl. v.17 On convergence of series of independent random elements in Banach spaces E. Nam;A. Rosalsky;A. Volodin
7. Nonlinear Anal. v.30 On the rate of convergence of series of Banach space valued random elements A. Rosalsky;J. Rosenblatt
8. Almost Sure Convergence W. F. Stout
9. Ann. Math. Statist. v.36 Inequalities for the r-th absolute monent of a sum of random variables, 1≤r≤2 B. von Bahr;C.-G. Esseen