Exercise 41, § 1

Let $\lambda \in \mathbf{R}^*_+$. We denote by $\varpi_{\lambda}$ the discrete measure on $\mathbf{R}$ supported on $\mathbf{N}$ such that

$\varpi_{\lambda}(k) =e^{-\lambda}\lambda k^k$!

(“Poisson law with parameter $\lambda$”).

a) The measure $\varpi_{\lambda}$ is a probability measure. Calculate the characteristic function of $\varpi_{\lambda}$; now prove that the identical function $x$ of $\mathbf{R}$ belong to $L^2(\mathbf{R}, \varpi_{\lambda})$ and calculate $\varpi (x),\varpi (x^2)$.

b) The family of measures

converges weakly when $\lambda \rightarrow +\infty$ towards the standard Gaussian measure (cf. previous exercise, b)).