6. Suppose that {Xn} is a sequence of random variables such that Xn +d X where E(X) is finite. We would like to investig

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6 Suppose That Xn Is A Sequence Of Random Variables Such That Xn D X Where E X Is Finite We Would Like To Investig 1 (118.36 KiB) Viewed 20 times

6. Suppose that {Xn} is a sequence of random variables such that Xn +d X where E(X) is finite. We would like to investigate sufficient conditions under which E(Xn) + E(X) (assuming that E(Xn) is well-defined). Note that in Theorem 3.5 of Knight's book, we indicated that this convergence holds if the Xn's are uniformly bounded. (a) Let 8 > 0. Show that E(JXn|1+0) = (1+0) = » [*x*P(|X.] > )dx (b) Show that for any M > 0 and 5 > 0 M " P(|Xn| > x)dx < E(|Xn]) M = P(X.) > sedu + 1 M8 * $***P(X\/> x)de M Again let s > 0 and suppose that E(|Xn|1+0) S K < for all n. Assuming that Xn +d X, use the results of parts (a) and (b) to show that E(|Xn]) + E([X]) and E(Xn) + E(X) as n +0. Hint: Use the fact that M 5. |1P(X./ > 2) – P(X/> x) dx +0 as n – for each finiteM.