Question: Embedded jump chain. Let E be countable and .Xn0 a Markov chain on E with transition matrix . Let T0 D 0 and Tk D

Embedded jump chain. Let E be countable and .Xn0 a Markov chain on E with transition matrix …. Let T0 D 0 and Tk D inf¹n > Tk1 W Xn ¤ Xn1º be the time of the kth jump of .Xn0. Show that the sequence X

k WD XTk , k 0, is a Markov chain with transition matrix

(x, y)/(1-(x,x)) if yx, I* (x, y) = 0 otherwise.

Show further that, conditional on .X
k /k0, the differences TkC1Tk 1 are independent and geometrically distributed with parameter 1 ….X
k;X
k /.

(x, y)/(1-(x,x)) if yx, I* (x, y) = 0 otherwise.

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