product of n unifom distribution as n-> infinty

Queshion:

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

Suppose are iid random variable each distributed U[1,0] (uniform distribution) Suppose 0 < a < b <: Show that:

tends to a limit as n tends to infinty and find an expression for it

Attempt at solution

take natural logs

let

let

let Moment generating function of W and Y = and respectively

by independence.

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

as MGF converges => distribuiton converges

the anwser doesn't seem right