Proving the limits of expectations, w/ E(U_i) = mu < infinity

palmtree

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I have the following questions.



Suppose U1, U2,..., are independent and identically distributed random variables with:

. . . . .\(\displaystyle E(U_i)\, =\, \mu\, <\, \infty\)

Show the following (assume f is a bounded continuous function):

. . .\(\displaystyle \displaystyle \mbox{(i) }\, \lim_{n\rightarrow \infty}\, Ef\left(\dfrac{1}{n}\, \sum_{i=1}^n\, U_i\right)\, =\, f(\mu)\)

. . .\(\displaystyle \displaystyle \mbox{(ii) }\, \lim_{n\rightarrow \infty}\, \int_0^{\infty}\, ...\, \int_0^{\infty}\, \mbox{exp}\left\{-\dfrac{1}{n^2}\, \sum_{i=1}^n\, u_i^2\right\}\, \mbox{exp}\left\{-(u_1\, +\, ...\, +\, u_n)\right\}\, du_1\, ...\, du_n\, =\, \dfrac{1}{e}\)



Can anyone show me how to prove the 2 parts? Thanks.
 

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