Let {j}, {j} and {Zj}, j = 1,0,1, be three doubly infinite, mutually independent sequences

Let {ϵj}, {ϵ′j} and {Zj}, j = … − 1,0,1,… be three doubly infinite, mutually independent sequences of independent unit exponential random variables. The bivariate sequence {Xj

,Yj}, j = … − 1,0,1,…, defined by is known as a bivariate exponential recurrence process. Show that

(i) Xj and Yj are unit exponential random variables.

(ii) cov(Xj

,Xj+h) = 2−h where h ≥ 0.

(iii) Xi is independent of Yj

.

(iv) Xi is independent of the sequence {Yj}.

Hence deduce that all third-order mixed cumulants involving both Xs and Ys are zero.

Show also that cum (Xj

, Xj+1, Yj

, Yj+1) = 1/12

(McCullagh, 1984c).