Suppose I have a positive integrable random variable $X$ s.t. $$E[e^X]=+\infty$$ Now let's take a series with general term $p_n$, summing to one, and define $$Z=\sum_{n>0}p_ne^{X_n}$$ and $U=\ln Z$ where $X_n$ are i.i.d. copies of $X$.
Now I have two questions :
- I think that $$\mathcal{L}_U(\lambda)=\cases{+\infty & if $\lambda>0$ \\ 1 & otherwise}$$ is that true? (Here $\mathcal{L}_U (\lambda)$ is the Laplace transform of $U$ at $\lambda$). My proof here is a fraud I think and before giving it I would like to see other people's ideas.
- If 1 is true can I deduce from this fact that $U$ is not integrable, and why? (Here I hope it's true)
Best regards