# Convergence of moment-generating function of weighted sums of random variables

This is a continuation of my earlier question. Once again, let $c_n$ be a sequence of positive real numbers such that $$\sum^{\infty}_{n=1}{c_n} = \infty, \qquad \sum^{\infty}_{n=1}{c_n^2} < \infty.$$ Let $X_n$ be a family of i.i.d random variables with $\mathbb{E}(X_n) = 0$ and $\sigma^2(X_n) = 1$ for each $n$, and define the random variable $$X = \sum^{\infty}_{n=1}{c_n X_n}.$$ Is it true that the moment-generating function $\mathbb{E}(e^{tX})$ exists and is equal to $\prod^{\infty}_{n=1}{\mathbb{E}(e^{c_n t X_n})}$? This seems to be a bit trickier than proving the corresponding question for the characteristic function.

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Your hypothesis does not ensure that $E(\text{e}^{tX_n})$ exists (meaning, is finite), for any real number $t\ne0$. – Did Aug 5 '11 at 17:50
Hence, since the $X_n$ are independent, it does not ensure either that $E(\text{e}^{tX})$ is finite. Does this answer your question? – Did Aug 11 '11 at 9:51

As pointed out by Did, it is possible that $\mathbb E[e^{tX_n}]=+\infty$ for $t\neq 0$ (the assumption of finiteness of the second moment is not enough). Therefore, by independence, we may have $\mathbb E[e^{tX}]=+\infty$ for $t\neq 0$.