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hw 3 - n = 0 1 2 Between time n and time n 1 each of these...

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205B homework, week 4; due Thursday February 19 Durrett Chapter 5 Exercises 5.7, 5.8, 5.11 1. Let ( X n ) be an irreducible Markov chain on S with transition matrix ( p ( x,y )). Let B be a ﬁnite subset of S such that the chain a.s. visits B inﬁnitely often. Let ( Z m ) be the chain watched only on B . Then Z is irreducible, and so has stationary distribution ˆ π , say. Deﬁne μ ( x,y ) = E x X n =0 1 ( X n = y,T B >n ) , x B, y S. π ( y ) = X x B ˆ π ( x ) μ ( x,y ) . Show that π is invariant, in the sense π ( y ) = X z S π ( z ) p ( z,y ) ≤ ∞ , y S. 2. A population consists of X n individuals at times
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Unformatted text preview: n = 0 , 1 , 2 ,... . Between time n and time n + 1 each of these individuals dies with probability p independently of the others; and the population at time n + 1 consists of the survivors together with an independent random (Poisson ( λ )) number of immigrants. Let X have arbitrary distribution. What happens to the distribution of X n as n → ∞ ? [Hint: consider ﬁrst the case where X has Poisson ( λ ) distribution] 3...
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