Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 27K (Probability and Measure)

For $d=1$ and $X\sim N(0,{\sigma }^2)$, the characteristic function isFor a multivariate normal distribution with covariance matrix ${\sigma }^2{I}_d$, the coordinates are uncorrelated and hence independent by independence of uncorrelated jointly normal variables. ThereforeThis is the characteristic function of an isotropic Gaussian vector.

For an orthogonal matrix $A$, invariance in distribution givesThe orthogonal group is transitive on each Euclidean sphere, so ${\phi }_X(u)$ depends only on $\Vert u{\Vert }^2$. DefineThe continuity of a characteristic function makes $f$ continuous. Taking $A=-I$ also gives $X\stackrel{d}{=}-X$, and henceso $f$ is real-valued. Thusas for every orthogonally invariant random vector.

Independence of the coordinates gives factorization of their joint characteristic function of a random vector. Orthogonal invariance under coordinate permutations makes all coordinate marginals identical. Hence, for $r_1,r_2\ge 0$ and $d\ge 2$,Therefore

First, $f(0)=1$. For every $r\ge 0$,It cannot vanish: if $f(r_0)=0$ for some $r_0>0$, thenwould imply $f(r_0/n)=0$ for every positive integer $n$, contradicting continuity at zero. Thus $f(r)>0$. Since every characteristic function has modulus at most one,

Now set $h(r)=-\log f(r)$. It is continuous, nonnegative, and additive:The classification of a continuous additive function on the nonnegative real numbers gives $h(r)=\alpha r$, where $\alpha =h(1)\ge 0$. Consequently

Combining the preceding parts,This is the characteristic function of $N(0,2\alpha I_d)$. The uniqueness theorem for characteristic functions therefore givesWhen $\alpha =0$, this means the degenerate distribution concentrated at the origin. This proves the Maxwell characterization of the normal distribution.