Question: 3. (4' = 2' + 2') Let X1, . .., Xn be i.i.d. random variables with E(X,) = / and Var(Xi) = 02 (both less

3. (4' = 2' + 2') Let X1, . .., Xn be i.i.d. random variables with E(X,) = / and Var(Xi) = 02 (both less than oo). We know from Khintchine's LLN that Xn(=), Xit +X.) , u. Now define Zn such that Zn = Xn with probability 1 - En an with probability En whereby En and an are constants satisfying that when n + 0o, En -+ 0 and Endn -+ 0o. a). Show that Zn H (Hint: You only need to prove Zn " Xn; with the fact that X, + p. the result follows). b). Does E( Zn - #) converge to zero? Show why or why not

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