ABSTRACT
The central problem of probability theory was to discover conditions under which distributions for sums of independent random variables would converge to the normal distribution. Very general sufficient conditions for this convergence were found by A. M. Liapounov. This chapter deals with the new direction of research, summation of a random number of random variables. It considers some properties of infinitely divisible distributions. The investigations of recent years have shown that infinitely divisible distributions play an important role in varied problems of probability theory. In particular, the class of limit distributions for sums of independent random variables has been found to coincide with the class of infinitely divisible distributions. It has been shown that if a sequence of infinitely divisible distributions converges to a limit distribution law, then this limit distribution is itself infinitely divisible. The chapter presents conditions which are sufficient for the convergence of a given sequence of infinitely divisible distribution functions to the limit.
