Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in \mathbb R^n$ and $\langle X,a\rangle(\omega)=\sum_{j=1}^{n}X_{j}(\omega)a_{j}$.
Suppose $X=(X_{1},X_{2},\cdots,X_{n})$, where each $\{X_{i}:1\le i\le n\}$ is a real valued random variable on $(\Omega ,\mathcal B,P)$ , Then show that $\{X_{i}:1\le i\le n\}$ are independent if and only if $$\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$$
where $a_{i}\in \mathbb R$, for $1\le i \le n$.
thanks for help
