Let $\mathbf{X}=\left(X_{1}, \ldots, X_{d}\right)$ be an $\mathbb{R}^{d}$-valued random variable. Given $u=\left(u_{1}, \ldots, u_{d}\right) \in$ $\mathbb{R}^{d}$ we let

$$\phi_{\mathbf{X}}(u)=\mathbb{E}\left(e^{i\langle u, \mathbf{X}\rangle}\right)$$

be its characteristic function, where $\langle\cdot, \cdot\rangle$ is the usual inner product on $\mathbb{R}^{d}$.

(a) Suppose $\mathbf{X}$ is a Gaussian vector with mean 0 and covariance matrix $\sigma^{2} I_{d}$, where $\sigma>0$ and $I_{d}$ is the $d \times d$ identity matrix. What is the formula for the characteristic function $\phi_{\mathbf{X}}$ in the case $d=1$ ? Derive from it a formula for $\phi_{\mathbf{X}}$ in the case $d \geqslant 2$.

(b) We now no longer assume that $\mathbf{X}$ is necessarily a Gaussian vector. Instead we assume that the $X_{i}$ 's are independent random variables and that the random vector $A \mathbf{X}$ has the same law as $\mathbf{X}$ for every orthogonal matrix $A$. Furthermore we assume that $d \geqslant 2$.

(i) Show that there exists a continuous function $f:[0,+\infty) \rightarrow \mathbb{R}$ such that

$$\phi_{\mathbf{X}}(u)=f\left(u_{1}^{2}+\ldots+u_{d}^{2}\right)$$

[You may use the fact that for every two vectors $u, v \in \mathbb{R}^{d}$ such that $\langle u, u\rangle=\langle v, v\rangle$ there is an orthogonal matrix $A$ such that $A u=v$. ]

(ii) Show that for all $r_{1}, r_{2} \geqslant 0$

$$f\left(r_{1}+r_{2}\right)=f\left(r_{1}\right) f\left(r_{2}\right) .$$

(iii) Deduce that $f$ takes values in $(0,1]$, and furthermore that there exists $\alpha \geqslant 0$ such that $f(r)=e^{-r \alpha}$, for all $r \geqslant 0$.

(iv) What must be the law of $\mathbf{X}$ ?

[Standard properties of characteristic functions from the course may be used without proof if clearly stated.]