N n0 to illustrate the use of kolmogorovs equations

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Unformatted text preview: it is not hard to fill in the missing details. Pn P1 Proof. var (Tn ) = m=1 1/m2 2 C = m=1 1/m2 2 . Chebyshev’s inequality implies P (Tn ETn /2) 4C/(ETn )2 ! 0 as n ! 1. Since n ! Tn is increasing, it follows that Tn ! 1. Our final example justifies the remark we made before Example 4.1. Example 4.6. Uniformization. Suppose that ⇤ = supi u(i, j ) = q (i, j )/⇤ u(i, i) = 1 i /⇤ i < 1 and let for j 6= i In words, each site attempts jumps at rate ⇤ but stays put with probability 1 i /⇤ so that the rate of leaving state i is i . If we let Yn be a Markov chain with transition probability u(i, j ) and N (t) be a Poisson process with rate ⇤ then Xt = YN (t) has the desired transition rates. This construction is useful because Yn is simpler to simulate that X (t) and has the same stationary distribution. 4.2 Computing the Transition Probability In the last section we saw that given jump rates q (i, j ) we can construct a Markov chain that has these jump rates. This chain,...
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