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Proof of the Weak Law of Large numbers by using Moment Generating Functions
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Jan2-12, 08:37 AM
1. The problem statement, all variables and given/known data
I need a thorough proof of the weak law of large numbers and it must use moment generating functions as below.
2. Relevant equations
The weak law of large numbers states that given X1.....Xn independent and identically distributed random variables with mean μ and variance σ2 then
X = (1/n) * Ʃ Xi tends to μ in distribution as n -> ∞
I am required to start with showing E[eθX ] → eθμ as n→∞
3. The attempt at a solution
This is what I have done.
E[eθX ] = E[eθ*(1/n) * Ʃ Xi ]
E[eθ*(1/n) * Ʃ Xi ] = Product of E[eθ*(1/n)*Xi ] from i = 1 to n
Since the random variables Xi are independent and identically distributed i can just consider the moment generating function of X1,
I know that [itex]\varphi[/itex]x1 (θ/n) = (1 + θμ/n + E[X12]θ2/n2)
By the taylor expansion of mgf up to order 1
E[eθ*(1/n) * Ʃ Xi ] = (1 + θμ/n + E[X12]θ2/n2)n
And so Log( E[eθ*(1/n) * Ʃ Xi ]) = nLog((1 + θμ/n + E[X12]θ2/n2))
= n(θμ/n + E[X12]θ2/n2)
By using Log(1+x) = x-x2/2 ...
= θμ + (E[X12]θ2/n ) → θμ as n→∞
Is this correct?
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