Explosion in finite time. Without the assumption (iii) in Problem 6.28, a Markov chain with generator G does not exist in general. For instance, suppose E D N and G.x; xC1/ D G.x; x/ D x2 for all x 2 N. Determine the discrete jump chain .Z

n/n0 with starting point 0 and jump times .T 

n /n1 as in Problem 6.28c and show that E0.supn1 T 

n/ < 1, i.e., the Markov chain .Xt /t0 ‘explodes’ at the almost surely finite time supn1 T 

n .