# Show convergence of product of n uniform distributions

1. Nov 11, 2009

### rosh300

1. The problem statement, all variables and given/known data
Suppose $$X_1, X_2, \ldots X_n$$ are iid random variable each distributed U[1,0] (uniform distribution) Suppose 0 < a < b <: Show that:
$$P((X_1X_2 \ldots \X_n)^{\frac{1}{\sqrt{n}}} \in [a,b])$$ tends to a limit as n tends to infinty and find an expression for it

2. Relevant equations

3. The attempt at a solution
$$P[log(a) \leq (X_1, X_2 \ldots X_n)^{\frac{1}{\sqrt{n}}} \leq b]$$
take natural logs
$$= P[log(a) \leq \frac{1}{\sqrt{n}}log(X_1, X_2 \ldots, X_n) \leq log(b)]$$

let $$Y = \frac{1}{\sqrt{n}}log(X_i) \Rightarrow f_y(x) = \sqrt{n}e^{\sqrt{n}y} \mbox{ for } x \in [0, log[0] 0 \mbox{otherwise}$$
let $$W~\frac{1}{\sqrt{n}}(log[X_1], log[X_2], \ldots log[x_n]) W ~ \Sigma^n_{i = 1} (Y_i)$$

let Moment generating function of W and Y = $$\Phi_{y_i}(t)$$ and $$\Phi_w(t)$$ respectively

$$\Phi_{w} = \Pi^n_{i=0}(\Phi_{y_i}(t))$$ by independence.

$$\Phi_{y} = \mathbb{E}[e^{xt}] = \frac{\sqrt{n}}{t + \sqrt{n}}$$

$$\Phi_{w} = (\frac{\sqrt{n}}{t + \sqrt{n}})^n$$

$$\stackrel{Lim}{n \rightarrow \infty}[ (\frac{\sqrt{n}}{t + \sqrt{n}})^n ] = e^{-2t} = e^{t\frac{1}{2}}$$
Converges to MGF of degenerate distribution with parameter 1/2.
as MGF converges => distribuiton converges

the anwser doesn't seem right

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