Extreme-value Process. Suppose Yn are i.i.d. integer-valued random variables with distribution pk = P{Y 1 = k}, where this probability is positive for some k > 0. Define X0 = 0 and Xn = max{Y 1 ,...,Y...

Extreme-value Process. Suppose Yn are i.i.d. integer-valued random variables with distribution pk = P{Y₁
= k}, where this probability is positive for some k > 0. Define X0 = 0 and
_Xn
= max{Y₁,...,Y_n}. Let τ0 = 0 and
_τn+1
= min{m>τn|X_m
> X_τn
}, n ≥ 0. Then Xn = X_τn
is the nth record value. Justify that
_Xn
and Xn are Markov chains and specify their transition probabilities. In addition, classify their states