RELATIVE ENTROPY UNDER MAPPINGS BY STOCHASTIC MATRICES

The relative g-entropy of two finite, discrete probability distributions x = (x1, ..., x(n)) and y = (y1, ..., y(n)) is defined as H(g)(x, y) - SIGMA(k)x(k)g(y(k)/x(k) - 1), where g : (- 1, infinity) --> R is convex and g(0) = 0. When g(t) = - log(1 + t), then H(g)(x, y) = SIGMA(k)x(k) log(x(k)/y(k)), the usual relative entropy. Let P(n) = {x is-an-element-of R(n) : SIGMA(i)x(i) = 1, x(i) > 0 for-all i). Our major result is that, for any m X n column-stochastic matrix A, the contraction coefficient defined as eta(g)(A) = sup{H(g)(Ax, Ay)/H(g)(x, y): x, y is-an-element-of P(n), x not-equal y) satisfies eta(g)(A) less-than-or-equal-to 1 - alpha(A), where alpha(A) = min(j, k) SIGMA(i) min(a(ij), a(ik)) is Dobrushin's coefficient of ergodicity. Consequently, eta(g)(A) < 1 if and only if A is scrambling. Upper and lower bounds on eta(g)(A) are established. Analogous results hold for Markov chains in continuous time.