Rates of convergence for classes of functions: The non-i.i.d. case

Let Xi, i >= 1, be a sequence of [phi]-mixing random variables with values in a sample space (X, A). Let L(Xi) = P(i) for all i >= 1 and let n, n >= 1, be classes of real-valued measurable functions on (X, A). Given any function g on (X, A), let Sn(g) = [Sigma]i = 1n {g(Xi) - Eg(Xi)}. Under weak metric entropy conditions on n and under growth conditions on both the mixing coefficients and the maximal variance V := V(n) := maxi <= n supg [set membership, variant] n [integral operator] g2 dP(i), we show that there is a numerical constant U < [infinity] such that a.s. *, where := [circle times operator]i = 1xP(i) and H := H(n) is the square root of the entropy of the class n. Additionally, the rate of convergence H-1(n/V)1/2 cannot, in general, be improved upon. Applications of this result are considered.