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Problems in Markov chains - ku

Problems inMarkov chainsDepartment of Mathematical SciencesUniversity of CopenhagenApril 2008 This collection of Problems was compiled for the course Statistik 1B. It con-tains the Problems in Martin Jacobsen and Niels Keiding: Markovk der(KUIMS 1990), and 1990 Torben MartinussenJan W. NielsenJesper MadsenIn this edition a few misprints have been correctedSeptember 1992S ren Feodor NielsenIn this edition a few misprints have been correctedSeptember 1993 Anders BrixTranslated into English. A number of additional Problems have been 2007 Merete Grove JacobsenS ren Feodor NielsenMisprints corrected and additional Problems 2008S ren Feodor Nielsen21. Conditional independenceProblem that there are functions (of sets)fzandgzsuch thatfor all setsAandBwe haveP{X A, Y B|Z=z}=fz(A)gz(B)for everyz. Show thatXandYare conditionally independent the result of the previous problem to show, or showdirectly, that ifP{X=x, Y=y, Z=z}=f(x, z) g(y, z)for some functionsfandgthenXandYare conditionally independent thatXandYare conditionally independent givenZifand only ifP{X A|Y=y, Z=z}=P{X A|Z=z}for every (measurable) setAand ((Y, Z)(P)-almost) every (y, z).

2. Discrete time homogeneous Markov chains. Problem 2.1 (Random Walks). Let Y0,Y1,... be a sequence of independent, identically distributed random variables on Z. Let Xn = Xn j=0 Yj n = 0,1,... Show that {Xn}n≥0 is a homogeneous Markov chain. Problem 2.2 Let Y0,Y1,... be a sequence of independent, identically dis- tributed random variables on N0.Let X0 = Y0 and

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