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8 Exercises
2. Let X1;X2;X3; ::: be independent r.v.'s such that P(X1 = 0) = P(X2 = 0) = 1 and
P(Xn = n) = P(Xn = ¡n) =
1
2n ln n
; P(Xn = 0) = 1 ¡
1
n ln n
; n ¸ 3:
Show that this sequence obeys the WLLN but not the SLLN. In other words, ¹X
!p 0, but ¹X
6!a:s: 0.
4. Let fXng be i.i.d. r.v.'s, and fCn; n ¸ 1g is a bounded sequence. Assume that EX1 = 0. Show
that
1
n
Xn
j=1
CjXj ¡! 0 a:s:
(Hint: Use truncation.)
5. Show that if X1;X2; ::: are independent with EXn = 0 and
1X
n=1
E
¡
X2n
IfjXnj · 1g + jXnjIfjXnj > 1g
¢
< 1;
then
P1
n=1 Xn converges a.s.
6. X1;X2; ::: are independent r.v.'s. Suppose that
1X
n=1
EjXnjpn < 1;
where 0 < pn · 2 for all n and EXn = 0 when pn > 1. Show that
P1
n=1 Xn converges a.s.
8. Let fXng be independent r.v.'s with
P(Xn = n®) = P(Xn = ¡n®) = 1=2; n = 1; 2; ::::::
Show that fXng satis¯es the SLLN if and only if ® < 1=2.

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