On the Asymptotic Distribution of the Sum of a Random Number of Random Variables.

then under appropriate conditions on the Xj it follows from the central limit theorem that the distribution of F will be nearly normal. In many cases of practical importance, however, the number N is itself a r. v., and when this is so the situation is more complex. We shall consider the case in which the Xj (j = 1, 2, • • • ) are independent r. v.'s with the same distribution function (d. f.) F(x) — P\Xj-£x\y and in which the non-negative integer-valued r. v. N is independent of the Xj. The d. f. of N we shall assume to depend on a parameter X, so that the d. f. of F is a function of X which may have an asymptotic expression as X—»oo. In the degenerate case in which for any integer X, N is certain to have the value X, the problem reduces to the ordinary central limit problem for equi-distributed components. In the general case the d. f. of N for any X is determined by the values cOjfc = P[iV = fe] (fe = 0, 1, • • • ), where the co& are f unctions of X such that for all X,