Exercise 6.9: Bounded Matrices and Bernstein Condition

chapter 6

(a)

Follows immediately from (b).

(b)

If $B - A \ge 0$, then clearly $Q(B - A)Q \ge 0$.

(c)

Use (a) and (b) to compute \begin{equation} \E[Q^j] = \E[Q Q^{j - 2} Q] \le b^{j-2} \E[Q^2] = b^{j-2} \Var(Q). \end{equation}