# Thread: Proving the limits of expectations

1. ## Proving the limits of expectations

 Hi, I have the following questions. Can anyone show me how to solve the 2 parts? i think it involves using Strong Laws of Large Numbers, Dominated Convergence Theorem, etc. For part (ii), it think the left-hand side can be expressed in the form of an expectation (similar to part (i)). ..

2. ## Re: Proving the limits of expectations

for (1) you use the Central Limit Theorem to push the limit inside the expectation

$\displaystyle \lim_{n\to \infty}E\left[f\left(\dfrac 1 n \sum_{k=1}^n~U_k\right)\right]=f\left(E\left[\lim_{n\to \infty}\dfrac 1 n \sum_{k=1}^n~U_k\right]\right) =f\left(E[N(\mu,0)]\right)=f(\mu)$

You should be more rigorous about the limit of the sample average converging to a Normal rv with mean $\mu$

And you should be more rigorous about the limit of a continuous bounded function of a variable being equal to the function of the limit of the variable.

I'll have to ponder the 2nd one a bit.