= Convergence theorem for moment-generating functions
Suppose the <moment-generating functions> $M_n$ are finite on a common open <interval> $(-a,a)$ containing zero and converge there pointwise to the <moment-generating function> $M_X$ of a <random variable> $X$, with $M_X$ finite there. Then $X_n$ tends to $X$ in <distribution>. Equivalently, a finite pointwise limit on such an interval is the <moment-generating function> of a unique limiting <probability distribution>. This theorem requires control on a neighborhood of zero; convergence of isolated <moments> is a different hypothesis.
Back to article page
