(Bennett's inequality) This exercise is devoted to a proof of a strengthening of Bernstein's inequality, known as Bennett's inequality. (a) Consider a zero-mean random variable such that IX; b for some b > 0. Prove that 108 E[et] 01 (0 (1 = 26} -1-Ab = for all R, (ab) where o var(X;). = (b) Given independent random variables X,..., X, satisfying the condition of part (a), let := be the average variance. Prove Bennett's inequality n P[X, 2 no] exp{-10 h (55)}. s X i=1 where h(t) = (1 +t) log(1 +t) - t for t > 0. (c) Show that Bennett's inequality is at least as good as Bernstein's inequality. (2.62)