(a) Consider a decision problem and decision rule & for which E[(L(0, 8(X))- R(0 8))] k

(a) Consider a decision problem and decision rule & for which E[(L(0, 8(X))- R(0 8))²] ≤k<∞∞ for all 0.

Imagine a sequence of repetitions of the decision problem, where the parameters 6, can change and the data X() (from the density f(x0,)) are independent. Show that, if R(0, 8) R for all 0, then 1 N lim L(0, 8(X())) R N N with probability one for any sequence (01, 02,...).

(1.2)

(b) Construct an example of a decision problem where sup, R{0, 8) = 00, and yet (1.2) still holds with probability one for any sequence (01, 02,...).