Is the Strong Law of Large Numbers applied on a series of non identically distributed exponential random variables?

Question

While working on a problem, I got stuck on this: I'm given a sequence of independent random variables ${X_n}$ such that for each $n\geq1$, $X_n\sim \exp(\lambda)$ for some fixed paramater $\lambda$. Then, we define a new series of random variables $W_n=\min\{X_1,\ldots,X_n\}$ and I need to show that the Strong Law applies to this Series.

I've shown that for each $n$, $W_n \sim\exp(\lambda n)$, and from that I can show that the series of the expectations converges to $0$, But I can't seem to be able to show the convergence a.s for the averages of the new r.v's. Will appreciate your help!

Answer 1

First of all...$W_n$ is a SEQUENCE, not a Series.

It is easy to verify that this sequence is monotonically decreasing

$$W_{n+1}=min[W_n;X_{n+1}] \leq W_n$$

and low-bounded to zero as $X_i \in [0:+\infty)$ for every $i$

Thus this sequence converges a.s. to zero

Answer 2

Use the Borel-Cantelli lemma and note that for every $\epsilon>0$, $$ \sum_{n\ge 1}\mathsf{P}(W_n>\epsilon)\le \sum_{n\ge 1}\frac{\mathsf{E}W_n^2}{\epsilon^2}= \sum_{n\ge 1}\frac{2}{(\lambda n\epsilon)^2}=\frac{1}{3}\left(\frac{\pi}{\lambda\epsilon}\right)^2<\infty. $$