On the Weak Law for Randomly Indexed Partial Sums for Arrays

Commun. Korean Math. Soc. 2001 Vol. 16, No. 2, 291-296

Dug Hun Hong, Soo Hak Sung, Andrei I. Volodin Catholic University of Taegu, Pai Chai University, Regina University

Abstract : For randomly indexed sums of the form $\sum_{i=1}^{N_n} (X_{ni}-c_{ni})/b_n,$ where $\{X_{ni}, i\ge 1, n\ge 1\}$ are random variables, $\{N_n, n\ge 1\}$ are positive integer-valued random variables, $\{c_{ni}, i\ge 1, n\ge 1\}$ are suitable conditional expectations and $\{b_n, n\ge 1\}$ are positive constants, we establish a general weak law of large numbers. Our result improves that of Hong [3].

Keywords : weak law of large numbers, convergence in probability, arrays, randomly indexed sums, martingale difference sequences