Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 27K / b / iv / Solution

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.