Question: 7.5 Let (Xn)n1 be i.i.d. random variables distributed as normal with mean 2 and variance 1. a) Show that the sequence Yn := 1 n

7.5 Let (Xn)n≥1 be i.i.d. random variables distributed as normal with mean 2 and variance 1.

a) Show that the sequence Yn := 1 n

n i=1 XieXi converges almost surely and in distribution and find the limiting distribution.

b) Answer the same question for the sequence Zn := X1 + ··· + Xn X2 1 + ··· + X2 n

.

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