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# Exsheet3 - (b X n → X and X n → Y almost surely Show...

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EXAMPLE SHEET 3 CLASSICAL PROBABILITY, 2008, J. MORIARTY 1. Assume that X n , n = 1 , 2 ,... are random variables on some proba- bility space such that X n c in distribution, where c is some constant. Show that X n c also in probability. 2. Prove the following implications: i) X n X a.s. Y n Y a.s. bracerightbigg = X n + Y n X + Y a.s. ii) X n X in r th mean Y n Y in r th mean bracerightbigg = X n + Y n X + Y in r th mean iii) X n X in probability Y n Y in probability bracerightbigg = X n + Y n X + Y in probability. 3. Assume that (a) X n X and X n Y , both in probability. Show that
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Unformatted text preview: (b) X n → X and X n → Y almost surely. Show that P ( X = Y ) = 1. (c) X n → X and X n → Y in r th mean for some r ≥ 1. Show that P ( X = Y ) = 1. 4. Give examples of two (non-constant) random variables X and Y such that (a) H¨older’s inequality reduces to an equality for some p > 1 and q > 1 (b) Minkowski’s inequality reduces to an equality for some p ≥ 1. 1...
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