6. Suppose that {X₂} is a sequence of random variables such that Xn →¿ Xwhere E(X) is finite. We would like to investiga

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6 Suppose That X Is A Sequence Of Random Variables Such That Xn Xwhere E X Is Finite We Would Like To Investiga 1 (118.36 KiB) Viewed 19 times

6. Suppose that {X₂} is a sequence of random variables such that Xn →¿ Xwhere E(X) is finite. We would like to investigate sufficient conditions under which E(Xn) → E(X) (assuming that E(X₂) 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 > 0. Show that E(|X₂|¹+6) = (1 + 6) * x*P(|X₂| > x)dx (b) Show that for any M > 0 and 8 > 0 M L™ P(|Xn| > x)dx ≤ E(|Xn]) M 1 f.M P(|Xn>x) dx - + M³ M xºP(|X₁| > x)dx Again let d > 0 and suppose that E(|X₂|¹+³) ≤ 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 → ∞. Hint: Use the fact that M ** |P(|X₂| > x) − P(|X| > x)| dx →0 as n→ for each finiteM.