Sequences of Random Variables

The basic concepts of probability theory, such as the probability of a random event (or the mean of a random variable), have been given intuitive meanings as approximately representing certain averages computed from a large sample of independent observed values of the event (or of the random variable). In this chapter we treat the problem of giving an exact mathematical meaning to the word “approximately” as it is employed in the foregoing sentence. At the same time, our discussion leads to an answer to the question of what constitutes an approximate solution to the problem of finding the probability law of the sum of random variables. A basic role in this study is played by the notion of the convergence of a sequence of random variables .