Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 24J / b / Solution

Independence gives ${\varphi }_{X-Y}(u)={\varphi }_X(u){\varphi }_X(-u)=|{\varphi }_X(u){|}^2=1$ for $|u|<\epsilon$. Taking real parts shows $\mathbb{E}[1-\cos (u(X-Y))]=0$. The integrand is nonnegative, so it vanishes almost surely for each such $u$.

Choose a countable sequence of positive $u_j\to 0$ in that interval, and intersect the corresponding probability-one events. On their intersection, a nonzero difference $D=X-Y$ would give $0<|u_{j}D|<2\pi$ for large $j$, contradicting $\cos (u_{j}D)=1$. Thus $X=Y$ almost surely, and part (a) proves $X$ is almost surely constant.