Exercise given a sequence of i.i.d. real RV and a distribution function $F$ need to verify three statements for $Z_n=n \min (X_1, \dots , X_n)$

Question

Pardon the title, I couldn't think of a better one. The three statements below are to prepare for a tougher exercise (which I could solve) however I am lost on 1 of the statements.

Problem: Let $(X_n)$ be a sequence of i.i.d. real RV with distribution function $F$ such that $F(t)/t \to \lambda>0$ as $t \to 0^+$. Define $Z_n:= n \min (X_1, \dots , X_n)$ and verify

1) For every $n \in \mathbb{N}, \ Z_n > 0$ almost surely
2) For every $t>0$, $P(Z_n >t) \to e^{- \lambda t}$ for $n \to \infty$
3) For every $\epsilon >0$ there is $n X_n \leq \epsilon$ inifinitely often, almost surely

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I managed to verify the first two. If you want to see my arguments to show my effort I will gladly write it down here. However they are really easy statements.

3) I am stuck here, the exercises cries Borel-Cantelli lemma. But I don't see how to apply it since I cannot find convergent/divergent bounds for $$P(nX_n \leq \epsilon) $$

Maybe someone can provide me with a hint to get me unstuck?

Answer 1

Borel Cantelli shows that the probability of events happening infinitely often is 0. You want to use the converse which says that if your events are independent, and the sum diverges then the occur infinitely often with probability 1.

$$\sum_{n} P(nX_n\leq \epsilon)=\sum_n P(X_n\leq \epsilon/n).$$

Now use the fact that

$$P(X_n\leq \epsilon/n)=F(\epsilon/n)=\frac{F(\epsilon/n)}{\epsilon/n}\frac{\epsilon}{n}\geq (\lambda-c(n))\frac{\epsilon}{n},$$

where you just need to pick $n$ large enough so that $c(n)$ is permanently smaller than $\lambda$.