Exam

If

P : E \to E

is a linear contraction, then

‖ e^{n (P - 1)} x - P^{n} x ‖ \leq \sqrt{n} ‖ (P - 1) x ‖

Let

N_{t}

be a Poisson process with parameter

λ = 1

. As

E N_{t} = t

and

E N_{t}^{2} = t^{2} + t

we infer that:

\begin{array}{rcl} ‖ e^{n (P - 1)} x - P^{n} x ‖ & \leq & E ‖ P^{N_{n}} x - P^{n} x ‖ \leq E | N_{n} - n | ‖ P x - x ‖ \\ \leq & (E (N_{n}^{2} + n^{2} - 2 n N_{n}))^{\frac{1}{2}} ‖ P x - x ‖ = \sqrt{n} ‖ P x - x ‖ . \end{array}