h1a - Probability II MAP6473 4810 Homework-1a Prof. JLF...

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Probability II MAP6473 4810 Homework-1a Prof. JLF King 4May2010 Reading. For Wednesday, 14Jan, please read Billingsley’s article “The Singular Function of Bold Play”. Please get your copy of his text as soon as pos- sible. Please read Sect.7, pages 92–108 by Monday, 19Jan. Please read part of Sect.8, pages 111–121 by Friday, 23Jan. 1a1: For a random variable Y define N ( Y ) := Z Ω Min( | Y | , 1) . Show that N ( Y ) = 0 ⇐⇒ Y a.e = 0 . Prove or disprove each of the following: i : N ( Y ) + N ( Z ) > N ( Y + Z ). ii : If N ( Y n ) 0 then ~ Y converges to 0 in proba- bility. iii : If ~ Y 0 in probability, then N ( Y n ) 0. Is convergence-in-probability convergence with re- spect to some metric? 1a2: Consider the following 3-state MC (Markov chain) with states A,B,C : State A goes to B and C each with prob= 1 2 . State B goes to states A,B,C with probabilities
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