(i) Show that Xn is uniformly integrable if and only if supn E Xn and sup n E Xn IA A Xn() d P() 0 as P A 0 (ii) Suppose X1, , Xn are i i d with finite mean Show that X n is uniformly integrable and hence E X n 0 (The fact that X n is uniformly integrable holds if the Xi are just identically distri...

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(i) Show that {Xn} is uniformly integrable if and only if supn E|Xn| < and sup...

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(i) Show that {Xn} is uniformly integrable if and only if supn E|Xn| < ∞ and sup n

E[|Xn|IA} =

|Xn(ω)|d P(ω) → 0 as P{A} → 0.

(ii) Suppose X1,..., Xn are i.i.d. with finite mean μ. Show that X¯ n is uniformly integrable and hence E|X¯ n − μ| → 0. (The fact that X¯ n is uniformly integrable holds if the Xi are just identically distributed with finite mean.)

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Testing Statistical Hypotheses Volume I

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Authors: E.L. Lehmann, Joseph P. Romano

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(i) Show that {Xn} is uniformly integrable if and only if supn E|Xn| < and sup...

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Testing Statistical Hypotheses Volume I

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