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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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