$\{X_i\}$'s are independent Poisson random variables with parameters $\lambda_i$ respectively satisfying $\sum_{n=1}^{\infty}\lambda_n=\infty$. Define $S_n=X_1+X_2+\cdots +X_n$ then show that $$\frac{S_n}{\mathbb{E}(S_n)}\to 1 \quad \text{a.e}$$
My try :
This looks like a problem with the method of subsequences. Let us first prove convergence in probability. $$P\left(|S_n-\mathbb{E}(S_n)|>\delta \mathbb{E}(S_n)\right)\le \frac{\text{var}(S_n)}{\delta^2(\mathbb{E}(S_n))^2}=\frac{1}{\delta^2\mathbb{E}(S_n)}\tag{$\ast$}$$ Since variance and mean of poisson are the same. Also the given series implies $\mathbb{E}(S_n)\to \infty$. So we have the probability convergence. Now define $n_k=\min \{n \mid \mathbb{E}(S_n)\geq k^2\}$ Now putting $T_k=S_{n_k}$ in $(\ast)$ we get that $$P\left(\left|\frac{T_k}{\mathbb{E}(T_k)}-1\right|>\delta\right)\leq \frac{1}{\delta^2k^2}$$ so summing over $k$, we get the lhs converges hence an easy application of Borel Cantelli Lemma 1 gives $\dfrac{T_k}{\mathbb{E}(T_k)}\to 1 \quad \text{a.e}$. Now I want to prove $$\frac{T_k}{\mathbb{E}(T_{k+1})}\le \frac{S_n}{\mathbb{E}(S_n)}\le \frac{T_{k+1}}{\mathbb{E}(T_{k})}$$ for $n_k\le n<n_{k+1}$ which would give me my solution. Which is obvious. But to finish the proof I need $\dfrac{\mathbb{E}(T_k)}{\mathbb{E}(T_{k+1})}\to 1$. But I can't show this fact. How to do this? I tried to do it with $k^2$ replaced by $2^k$, but it doesn't help. Can someone help?
EDIT : I found this pdf which solves this problem in page 3. There they claim, that if we show $I_n/\mathbb{E}(I_n)\to 1 \; \text{a.e}$ then they also have $S_n/\mathbb{E}(S_n)\to 1\; \text{a.e}$. Why is that? I can see the convergence in probability for $S_n$ but nothing further. Can someone help me? Thanks a lot.
You want to show $$ \frac{\operatorname{E}(T_k)}{\operatorname{E}(T_{k+1})} \to 1 \text{ as } k \to \infty. $$ You know $$ k^2 \le \operatorname{E} (T_k) <(k+1)^2 \le \operatorname{E}(T_{k+1}) < (k+2)^2. $$ Therefore $$ \frac{k^2}{(k+2)^2} \le \frac{\operatorname{E}(T_k)}{\operatorname{E}(T_{k+1})} \le 1. $$ You can probably go on from there.