Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.