Convergence of inverse of iid uniform random variable

Question

I am really struggling with the following problem.
Let
$$X_j $$ be iid random variable following $U[0,1]$.
Prove that $$
\lim_{n\to\infty}\frac{n}{X_1^{-1}+X_2^{-1}+…+X_n^{-1}}$$ exists almost surely and find its limit.

I really do not know how to apply the strong law of large numbers to the inverse of Uniform variables.
Any help, solution appreciated.

Answer 1

First set $Y=\frac{1}{X}$ and get that

$$f_Y(y)=\frac{1}{y^2}\cdot \mathbb{1}_{[1;+\infty)}(y)$$

With $\mathbb{E}[Y]=+\infty$

Thus using SLLN you get

$$\overline{Y}_n\xrightarrow{a.s.}\infty$$

and thus

$$\frac{n}{\Sigma_i Y_i}\xrightarrow{a.s.}0$$