24. Let X1, X2, ... be a sequence of nonnegative pairwise independent and identic- ally distributed random numbers with

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24 Let X1 X2 Be A Sequence Of Nonnegative Pairwise Independent And Identic Ally Distributed Random Numbers With 1 (56.21 KiB) Viewed 15 times

24. Let X1, X2, ... be a sequence of nonnegative pairwise independent and identic- ally distributed random numbers with mean EX = u <0. (a) Let Y, = X, x1(x,<n) be a truncated version of each Xn. Show that X,-7, 40. (b) By changing the order of summation, prove the following bound: È É BY1v Br? 14-1s5<) sep + x). Q BA (c) Let Bx = fak] for a > 1 be a subsequence of k = 1, 2,.... Show that Y8A-B7B0. (d) Use the result that if an + a, then IP-1 ax + a to show that EY B2 – . Hence, deduce that 7 H. (e) Show that for Bk sns Bk+1, we have: BK siis Bk+ly Bk+1 Bk Bk+ Hence, using all of the previous results deduce that X, u. YBK