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Show convergence of product of n uniform distributions

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose [tex] X_1, X_2, \ldots X_n [/tex] are iid random variable each distributed U[1,0] (uniform distribution) Suppose 0 < a < b <: Show that:
    [tex] P((X_1X_2 \ldots \X_n)^{\frac{1}{\sqrt{n}}} \in [a,b]) [/tex] tends to a limit as n tends to infinty and find an expression for it

    2. Relevant equations



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

    let [tex] 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} [/tex]
    let [tex] W~\frac{1}{\sqrt{n}}(log[X_1], log[X_2], \ldots log[x_n])
    W ~ \Sigma^n_{i = 1} (Y_i) [/tex]

    let Moment generating function of W and Y = [tex] \Phi_{y_i}(t) [/tex] and [tex] \Phi_w(t) [/tex] respectively

    [tex]\Phi_{w} = \Pi^n_{i=0}(\Phi_{y_i}(t)) [/tex] by independence.

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

    [tex] \Phi_{w} = (\frac{\sqrt{n}}{t + \sqrt{n}})^n[/tex]

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

    the anwser doesn't seem right
     
  2. jcsd
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