Exercise 39, § 1

Let $\nu \in \mathscr{P}(\mathbf{R})$ be a probability measure on $\mathbf{R}$. The characteristic function of $\nu$ is the function $\varphi_{\nu}$ from $\mathbf{R}$ into $\mathbf{C}$ such that

for every $x\in \mathbf{R}$.

a) For $\nu_1,\nu_2$ in $\mathscr{P}(\mathbf{R})$, one has $\nu_1=\nu_2$ if and only if $\varphi_{\nu_1}=\varphi_{\nu_2}$.

b) For $\nu_1,\nu_2$ in $\mathscr{P}(\mathbf{R})$, one has $\nu_1*\nu_2\in \mathscr{P}(\mathbf{R})$ and $\varphi_{\nu_1*\nu_2}=\varphi_{\nu_1}\varphi_{\nu_2}$.

c) Let $\nu \in \mathscr{P}(\mathbf{R})$. The function $\varphi_{\nu}$ is bounded by 1 and uniformly continuous on $\mathbf{R}$.

d) Let $\nu \in \mathscr{P}(\mathbf{R})$. For all real numbers $a < b$, one has

$\nu (]a, b[) +\frac{1}{2}\nu (\{a, b\}) =$ lim$_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^T\varphi_{\nu}(x)\frac{e^{-ixa}-e^{-ixb}}{ix}dx$,

and in particular the limit always exists. Moreover, for every $T>0$, one has

T $_{-T}$

e) Let $(\nu_n)_{n\in\mathbf{N}}$ be a sequence of probability measures on $\mathbf{R}$ converging narrowly (INT, IX, p. 59, déf. 1) to a measure $\nu$; then $\nu \in \mathscr{P}(\mathbf{R})$ and $\varphi_{\nu_n}(x)$ converges to $\varphi_{\nu}(x)$ for every $x\in \mathbf{R}$.

Let $(\nu_n)_{n\in\mathbf{N}}$ be a sequence of probability measures on $\mathbf{R}$. Suppose that for every $x\in \mathbf{R}$, the sequence $(\varphi_{\nu_n}(x))_n$ converges to a complex number $\varphi (x)$, and that the function $x\mapsto \varphi (x)$ is continuous at 0.

f) Let $\varepsilon >0$. There exists $a >0$ such that

for every integer $n\in \mathbf{N}$.

g) The sequence $(\nu_n)$ is relatively compact for the narrow topology.

h) The sequence $(\nu_n)$ converges narrowly to a probability measure $\nu$ whose $\varphi$ is the characteristic function (« Continuity theorem of P. Lévy »). i) There exist sequences $(\nu_n)$ of probability measures on $\mathbf{R}$, not converging narrowly to a probability measure, such that $\varphi_{\nu_n}(x)$ converges for every $x\in \mathbf{R}$.