2.8 Continue with Problem 2.7 with an = n. Show the following:

(i) If E(|X1|) < ∞, then ξn L1

−→ 0 as n→∞.

(ii) If E(X2 1) < ∞, then ξn

−a→.s. 0 as n→∞.

Hint: For (i), first show that for any a > 0, max 1≤i≤n

|Xi| ≤ a +

n i=1

|Xi |1(|Xi |>a).

For (ii), use Theorem 2.8 and also note that by exchanging the order of summation and expectation, one can show for any  > 0,

∞

n=1 nP(|X1| > n) < ∞.