Originally Posted by

**rosh3000** Queshion:

The problem statement, all variables and given/known data[/b]

Suppose $\displaystyle X_1, X_2, \ldots X_n $ are iid random variable each distributed U[1,0] (uniform distribution) Suppose 0 < a < b <: Show that:

$\displaystyle P((X_1X_2 \ldots X_n)^{\frac{1}{\sqrt{n}}} \in [a,b]) $ (LaTeX error corrected : \X_n instead of X_n) tends to a limit as n tends to infinty and find an expression for it

Attempt at solution

$\displaystyle P[a \leq (X_1, X_2 \ldots X_n)^{\frac{1}{\sqrt{n}}} \leq b] $

take natural logs

$\displaystyle = P[log(a) \leq \frac{1}{\sqrt{n}}log(X_1, X_2 \ldots, X_n) \leq log(b)]$

let $\displaystyle 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]) $ (problem with variable names, and sign in the exponent; nb: $\displaystyle \log(0)=-\infty$

let $\displaystyle W = \frac{1}{\sqrt{n}}(log[X_1] + log[X_2], \ldots + log[x_n]) $

$\displaystyle W = \Sigma^n_{i = 1} (Y_i) $

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

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

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

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

$\displaystyle \stackrel{Lim}{n \rightarrow \infty}[ (\frac{\sqrt{n}}{t + \sqrt{n}})^n ] = e^{-2t} = e^{t\frac{1}{2}} $ Wrong

Converges to MGF of degenerate distribution with parameter 1/2.

as MGF converges => distribuiton converges

the anwser doesn't seem right