# 8. X1,...,Xn are i.i.d. r.v.’s with μ = EX1 and EXi &l...

## Transcribed Text

8. X1,...,Xn are i.i.d. r.v.’s with μ = EX1 and EXi < ∞. Let Sn = 1 Xi. Show that Sn →μ, a.s. n 10. Let {Xn} be i.i.d. r.v.’s. Then (a) n−1 max1≤k≤n |Xk| →p 0 ⇐⇒ nP(|X1| > n) = o(1). (b) n−1 max1≤k≤n |Xk| → 0 a.s. ⇐⇒ E|X1| < ∞. Then apply the Dominated Convergence Theorem to Yn since E(|X| + |Z| + 2ε) < ∞. 18. Give an example to illustrate that Xn −→ X a.s. may not imply Xn ≤X+ε, for large enough. 20. Let X, X1, X2, ... be r.v.’s such that Xn −→a.s. X. Show that supn |Xn| = Op(1).

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