Solution to Problem 4.

(4) [20 points] Let $U_1$, $U_2$, $\ldots$ be i.i.d. uniform(0,1) rv's. Define

$$X_n \; = \; \left( \prod_{i=1}^n U_i \right)^{1/n} .$$

(a) Show that $X_n \rightarrow 1$ almost surely.

Solution: Using the hint, let $Y_i = - \log U_i$, then it is easy to check that $Y_i$ is exponential(1). Hence,

$$- \log X_n \; = \; \frac{1}{n} \sum_{i=1}^n Y_i \; \longrightarrow \; E[Y_i] = 1 , \quad \mbox{a.s.}$$

by the Strong Law of Large Numbers. Thus, taking $\exp[ - \cdot ]$ of both sides, we see that

$$X_n \; \longrightarrow \; e^{-1} , \quad \mbox{a.s.}$$

(b) Show that

$$\sqrt{n} \left( X_n - e^{-1} \right) \, \stackrel{D}{\rightarrow}\, N(0,1) .$$

Solution By the Central Limit Theorem,

$$\sqrt{n} \left[ - \log X_n - 1 \right] \; \stackrel{D}{\rightarrow}\; N(0, \mbox{Var} [Y_i] ) \; = \; N(0,1) .$$

Apply the $\delta$-method with $h(x) = \exp[- x]$, and noting that $h_{\prime} (1)^2 \; = \; e^{-2}$, we have

$$\sqrt{n} \left( X_n - e^{-1} \right) \; \stackrel{D}{\rightarrow}\; N(0,e^{-2}) .$$

Hint: logarithms.