Excursions from a recurrent state. Consider a Markov chain with a countable state space E and transition matrix … that starts in a recurrent state x 2 E. Let T0 D 0 and, for k  1, let Tk D inf¹n > Tk1 W Xn D xº be the time of the kth return to x and Lk D Tk  Tk1 the length of the kth ‘excursion’ from x. Show that, under Px, the random variables Lk are

(almost surely well-defined and) i.i.d.