Often minimax rates of convergence are defined using tail probabilities instead of expectations. For example, in Stone (1982) {an} is called the lower rate of convergence for the class D if...

Often minimax rates of convergence are defined using tail probabilities instead of expectations. For example, in Stone (1982) {an} is called the lower rate of convergence for the class D if

                       lim inf n→∞ inf mn sup (X,Y )∈D P mn − m2 an ≥ c  = C0 > 0. (3.7)

Show that (3.7) implies that {an} is a lower minimax rate of convergence according to Definition 3.1. Problem 3.5. Show that (3.7) holds for D = D(p,C) and an = n− 2p 2p+d .

Hint: Stone (1982).