Exercise 3.1 Consider the tent map series defined by Xi+1 = 1 − |2Xi − 1| =

2min(Xi, 1 − Xi), with X1 ∼ Unif(0, 1). Show that Xi ∼ Unif(0, 1) for all i ≥ 1. Also prove that the correlation between Xi and Xi+j is 0 for all i, j ≥ 1.

Hint: Xi+j is a function of |Xi−1/2| for all j ≥ 1, and i = sgn(Xi−1/2) =

 +1, Xi > 1/2

−1, Xi < 1/2

, is independent of |Xi − 1/2|.