This preview shows page 1. Sign up to view the full content.
Unformatted text preview: it is
not hard to ﬁll in the missing details.
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
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,...
View Full Document
- Spring '10
- The Land