Let $X$ be a $\mathbb{R}^d$-valued random variable. Denote by $\mathbb{P}_{X}$ its distribution and $\phi_{X}$ its characteristic function. I want to prove that the following are equivalent
- $\mathbb{P}_{-X} = \mathbb{P}_{X}$
- $\phi_{X} = \phi_{-X}$
- $\phi_{X}(t) = \mathbb{E}[\cos(\langle t, X \rangle]$ for all $t \in \mathbb{R}$
- $\phi_X \in \mathbb{R}$.
I am having trouble with finding a connection between $\phi$ and $\mathbb{P}$ and therefore can't prove any of the directions $(x) \implies (1)$, where $x \in \{2,3,4\}$.
I have proven the equivalence of the last three statements like this and would appreciate any feedback:
(3) $\implies$ (4). By definition we have \begin{equation*} \phi_{X}(t) \overset{\textrm{Def.}}{=} \mathbb{E}[\exp(i \langle t ,X \rangle)] \overset{\textrm{L}}{=} \mathbb{E}[\cos(\langle t ,X \rangle)] + i \mathbb{E}[\sin(\langle t ,X \rangle)] \overset{!}{=} \mathbb{E}[\cos(\langle t, X \rangle)]. \end{equation*} Therefore we have $i \mathbb{E}[\sin(\langle t ,X \rangle)] = 0$.
(4) $\implies$ (2). For all $a,b \in \mathbb{R}$ we have \begin{equation*} \phi_{a X + b}(t) = e^{i t b} \phi_{X}(at) \qquad \text{and} \qquad \phi_X(-t) = \overline{\phi_X(t)}. \end{equation*} This implies \begin{equation*} \phi_{-X}(t) = \phi_{X}(-t) = \overline{\phi_{X}(t)} \end{equation*}
(2) $\implies$ (3): By definition we shall have \begin{equation*} \mathbb{E}[\cos(\langle t, X \rangle)] + i \mathbb{E}[\sin(\langle t, X \rangle)] \mathbb{E}[\sin(\langle t, -X \rangle)], \end{equation*} which is equivalent to \begin{equation*} \mathbb{E}[\cos(\langle t, X \rangle)] + i \mathbb{E}[\sin(\langle t, X \rangle)] = \mathbb{E}[\cos(\langle t, X \rangle)] - i \mathbb{E}[\sin(\langle t, X \>)]. \end{equation*} This implies \begin{equation*} i \mathbb{E}[\sin(\langle t, X \rangle)] = - i \mathbb{E}[\sin(\langle t, X \rangle)] \implies i \mathbb{E}[\sin(\langle t, X \rangle)] = 0 \end{equation*}