7.13 Let Xn, n 1 be i.i.d. random variables with some distribution with finite mean and variance 2. For every n 1,

7.13 Let Xn, n ≥ 1 be i.i.d. random variables with some distribution with finite mean μ and variance σ2. For every n ≥ 1, we denote Zn := n i=1 eXi

 1 n

.

Find the almost sure limit of the sequence {Zn}n≥1.

7.14 Let {Xn}n≥1 be a sequence of i.i.d. random variables with common distribution N(μ, σ2), where μ ∈ R and σ > 0. For every n ≥ 1, denote Sn = n k=1 Xk.

a) Give the distribution of Sn.

b) Compute E



(Sn − nμ)

4

.

c) Show that for every ε > 0, P

!

Sn n − μ

≥ ε

"

≤

3σ4 n2 ε4 .

d) Derive that the sequence Sn n

n≥1 converges almost surely and identify the limit.

e) Do you recognize this result?