# Strong Law of Large Numbers and Uniformly Distributed Random Numbers

Let $$X_j$$ be i.i.d. $$\mathcal{U}[0,1]$$ random variables. Prove that $$\lim\limits_{n \to\infty} \frac{n}{X_1^{-1}+\dots+X_n^{-1}}$$ exists almost surely and find the limit.

I think I need to use the Strong Law of Large Numbers for this question. The statement of SLLN is stated as follows:

Let $$\{Y_j\}_{j=1}^\infty$$ be a sequence of i.i.d. random variables such that $$\mathbb{E}|Y_j| < \infty$$. Then $$\frac{S_n}{n} = \frac{Y_1+\dots+Y_n}{n}\to \mathbb{E}(Y_1)$$ almost surely.

Now, I did the following: Let $$Q = \frac{n}{X_1^{-1}+\dots+X_n^{-1}} \implies Q^{-1} = \frac{X_1^{-1}+\dots+X_n^{-1}}{n} = \frac{Y_1+\dots+Y_n}{n}$$ where $$Y_i = X_i^{-1}$$. However, $$\mathbb{E}(Y_i) = \int_0^1 \frac{1}{x}dx = +\infty$$

So I can't use the SLLN. But I really can't think of another approach or any useful transformation to deal with this question.

Thanks so much in advance for any insights.

Actually SLLN is valid for non-negative random variables when the mean is $$\infty$$: IF $$(Y_i)$$ is i.i.d. and non-negative then $$\frac {Y_1+Y_2+...+Y_n} n \to EY_1$$ almost surely whether or not $$EY_1<\infty$$.
This can be prove easily using a truncation argument. If $$Z_i=Y_iI_{Y_i \leq M}$$ then $$Y_i \geq Z_i$$ and SLLN applied to $$(Z_i)$$ shows that $$\lim \inf \frac {Y_1+Y_2+...+Y_n} n \geq EZ_1$$. Now let $$M \to \infty$$.
• Sorry, what is $M$?
• @xf16 $M$ is an arbitrary positive number. At the end you take limit as $M \to \infty$. Mar 26, 2020 at 5:07