The Method of Moments

The topic of this chapter is a basic method for proving convergence in distribution of a sequence of random variables \(X_n, n \geq 1\) , to a random variable X, the so-called method of moments. In this connection, a key notion is that of a uniformly integrable sequence of random variables. If the sequence \(X_1,X_2, \ldots \) is uniformly integrable, then convergence in distribution of \(X_n\) to X implies convergence \(E(X_n)\to E(X)\) of expectations. Suppose that for each integer k the kth moment of X and of \(X_n, n \geq 1\) , exists, and that the distribution of X is determined by the sequence, \(E(X^k),\ k \geq 1\) , of moments. The latter condition holds if the moment-generating function of X is finite in a neighborhood of 0. The method of moment states that, under this condition, \(X_n\) converges in distribution to X if \(E(X_n^k)\) converges to \(E(X^k)\) for each k. The normal distribution and the Poisson distribution are determined by the sequence of respective moments, but not the lognormal distribution.