Proof of the Weak Law of Large numbers by using Moment Generating Functions

[stukbv]

Homework Statement

I need a thorough proof of the weak law of large numbers and it must use moment generating functions as below.

Homework 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[e^θX ] → e^θμ as n→∞

The Attempt at a Solution

This is what I have done.

E[e^θX ] = E[e^{θ*(1/n) * Ʃ Xi} ]
E[e^{θ*(1/n) * Ʃ Xi} ] = Product of E[e^θ*(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 $\varphi$x1 (θ/n) = (1 + θμ/n + E[X1²]θ²/n²)
By the taylor expansion of mgf up to order 1

So now

E[e^{θ*(1/n) * Ʃ Xi} ] = (1 + θμ/n + E[X1²]θ²/n²)ⁿ

And so Log( E[e^{θ*(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?

 

[mighty2000]

Thank you for your question. Your approach to proving the weak law of large numbers using moment generating functions is a valid one. However, there are a few corrections and clarifications that I would like to make.

Firstly, when you write "Product of E[eθ*(1/n)*Xi ] from i = 1 to n," it should actually be the product of the moment generating functions, i.e. ΦXi (θ/n), not just the expectation values.

Secondly, the statement "By the taylor expansion of mgf up to order 1" is slightly misleading. The Taylor expansion of the moment generating function is up to infinite order, not just order 1. However, in this case, we only need to consider up to order 1 for our proof.

Thirdly, when you write "And so Log( E[eθ*(1/n) * Ʃ Xi ]) = nLog((1 + θμ/n + E[X12]θ2/n2))," it should be Log(ΦX (θ/n)), not Log(E[eθ*(1/n) * Ʃ Xi ]).

Lastly, your final expression should be θμ + (E[X12]θ2/n2), not just θμ, as you have correctly stated in the previous line.

Overall, your proof is on the right track, but there are some minor errors and clarifications that need to be made. I hope this helps and please let me know if you have any further questions.