CHAPTER IV - INFINITE SERIES OF RANDOM VARIABLES AND RELATED TOPICS

Abstract

This chapter defines various modes of stochastic convergence of the sequences of random variables. This enables the consideration of infinite series of random variables and to say that an infinite series of random variables is convergent (in the sense of a particular stochastic convergence concept) if the sequence of its partial sums converges. The chapter discusses the infinite series of random variables and related problems. A sequence of independent random variables is either almost certainly convergent or almost certainly divergent. If it is almost certainly convergent, then the limit is almost certainly a constant (a degenerate random variable). A series of independent random variables is either almost certainly convergent or almost certainly divergent and every sequence of independent random variables with uniformly bounded variances obeys the strong law of large numbers. A sequence of trials in a Bernoulli scheme obeys the strong law of large numbers, that is, the relative frequency of success converges almost certainly to the probability of success.