In wikipedia, it's written that there is a one-to-one correspondance between CDF and characteristic function of a r.v. Could someone give me this bijection ?
Moreover, what does it mean that "The characteristic function $\varphi _X$ determine uniquely the law of $X$" ? (written in the french wikipedia). Indeed, by the "bijection" between CDF and characteristic functions, I can imagine that knowing the characteristic function of the r.v. $X$, we can deduce the law of $X$. But what does it mean "uniquely" ? Because a r.v. can't have several law.
Essential is that - if $\mathcal F$ denotes the collection of all CDF's on $\mathbb R$ - the function $\chi:\mathcal F\to\mathbb C^{\mathbb R}$ prescribed by:$$F\mapsto\phi_F(t):=\int e^{itx}dF(x)$$ is injective. Here the RHS must be read as an element of $\mathbb C^{\mathbb R}$.
So if $\mathcal X$ is defined as the image of the function $\chi$ then automatically the restricted function $\mathcal F\to\mathcal X$ prescribed by $F\mapsto\chi(F)$ is a bijection.