03faex1 - 632 Introduction to Stochastic Processes Fall...

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632 Introduction to Stochastic Processes Fall 2003 Midterm Exam I Instructions: Show calculations and justify non-obvious statements for full credit. When you quote a result proved in class, state the hypotheses and conclusion clearly, and justify why the hypotheses are met. The points add up 100. General notation: P x ( A ) is the probability of the event A when the chain starts in state x , P μ ( A ) the probability when the initial state is random with distribution μ . T y = min { n 1 : X n = y } is the first time after 0 that the chain visits y , or if no visit to y ever happens. ρ x,y = P x ( T y < ) is the probability that the chain visits y some time after time 0, given that it started at x . N ( y ) is the number of visits to state y , not counting a possible visit at time 0. 1. For this problem, let P be defined by P = 0 1 0 0 0 1 1 / 3 2 / 3 0 . (a) (15 pts) Explain why the power P n of the matrix converges as n → ∞ and find the limit matrix. (b) (10 pts) Let
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03faex1 - 632 Introduction to Stochastic Processes Fall...

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