I am trying to solve the following problem:
Let $\lambda>0$ and $(X_n, n\geq 1)$ be a sequence of i.i.d. r.v. with $X_i\sim \mathcal{E}(\lambda)$.
Define as $Y_n= e^{X_n}, S_n = \sum_{j=1}^{n} Y_i$
a) For what values of $\lambda>0$ does there exist a number $\mu$ s.t. $\lim_{n\to +\infty} S_n/n \to \mu $ a.s.?
b) For what values of $\lambda>0$ the sequence of r.v. $$Z_n = \frac{S_n - n\mu}{\sqrt(n)}$$ converges in distribution to a limiting r.v. $Z$.
c) For what values of $\lambda>0$ there exists an $s_0>0$ such that $$\Lambda(s)=\log(\mathbb{E}(e^{sY_1}))<+\infty, \forall |s|\leq s_0$$
Here is my reasoning:
For a): the sequence of $Y_n$ is i.i.d. and the cdf of $Y_i$, $$F_{Y_i}(t)=F_x(\ln(t))=1-e^{\lambda \ln(t)} = 1-t^{-\lambda}, t>0$$
Being the seqence iid the sequence $S_n/n$ by the strong law of large numbers converges a.s. to $\mathbb{E}(Y_1)=\frac{\lambda}{1-\lambda}(e^{t(1-\lambda)})\big |^{+\infty}_0$ which is finite only if $\lambda>1$. Is that correct?
For b): using the Central Limit Theorem we have that $$\frac{Z_n}{\sigma} \to^d Z \sim \mathcal{N}(0,1)$$ where $\sigma = \sqrt(\operatorname{Var}(Y_i))$ thus, $Z_n \to^d \sigma Z \sim \mathcal{N}(0,\sigma^2)$ and $\sigma^2 = \operatorname{Var}(Y_i)$ which is finite as long as $\mathbb{E}(Y_i^2)<+\infty$ which happens if $\lambda>2$. So, we can say that $\operatorname{Var}(Y_i)= \frac{\lambda}{(\lambda-2)(\lambda-1)^2}$ and the converging distribution is $$Z' \sim \mathcal{N}(0,\operatorname{Var}(Y_i))$$ is that correct? I am not looking for (only) error in calculation but maninly in conceptual errors I might be making...
For c) I am not really sure on how to proceed, I tried to compute $\mathbb{E}(e^{sY_i})$ but I came across the integral $$\lambda\int_{0}^{+\infty} e^{sy}y^{-\lambda-2}$$ which I think should be solved by reiterated integration by parts until $\frac{\partial y^{-\lambda-2}}{\partial{y}}$ reaches $0$ is this the way to go? Or else, should I compute $\mathbb{E}(e^{s\dot s \dot X_1})$. I also tried to bound $\log(\mathbb{E}(e^{sY_1}))\leq \mathbb{E}(sY_1)$ using Jensen's inequality, but the last term seems to be always divergent and thus I cannot distinguish much. Any suggestions?
There is no $\lambda >0$ such that $Ee^{sY_1} <\infty$ for $|s|$ sufficiently small. In fact $se^{x}-\lambda x \to \infty$ as $x \to \infty$ whenever s$>0$.
As far as a) and b) are concerned you have proved one way implications. It can be shown that $S_n /n \to \infty$ a.s. in the case of non-negative random variables with infinite mean. Using this it follows that you answer to a) works in the opposite direction also: if $\lambda \leq 1$ then $S_n/n$ does not converge a.s. to a finite limit. Whether $Z_n /n$ can converge in distribution to a limiting variable even when the variance is not finite is a harder question.