Let S0 = 0 and Sn = Y 1 + ··· Y n , for n ≥ 1, where Yk arei.i.d. nonnegative integer valued random variables. Let Xn denote the unitin the integer expansion of Sn (so Xn equals Sn modulo 10; if Sn =...

Let S0 = 0 and Sn = Y₁
+ ··· Y_n, for n ≥ 1, where Yk arei.i.d. nonnegative integer valued random variables. Let Xn denote the unitin the integer expansion of Sn (so Xn equals Sn modulo 10; if Sn = 321,then Xn = 1). Show that Xn is a Markov chain and specify its transitionprobabilities. Under what conditions is Xn irreducible. Assuming it is, find itsstationary distribution. (Similar properties hold when X_n
equals Sn modulo10k for some k ≥ 1.)