Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 26G (Probability and Measure)

Lévy's convergence theorem says that probability measures ${\mu }_n$ on ${\mathbb{R}}^d$ converge weakly to $\mu$ if and only if their characteristic functions converge pointwise to the characteristic function of $\mu$. More generally, if the characteristic functions converge pointwise to a function continuous at zero, that function is the characteristic function of a probability measure and the corresponding weak convergence holds.

If $X_n\Rightarrow X$, the continuous mapping theorem givesfor every $u$. Conversely, the assumed one-dimensional convergence at $t=1$ givesfor every $u\in {\mathbb{R}}^d$. Lévy's convergence theorem then yields $X_n\Rightarrow X$. This is the Cramer-Wold theorem.

Fubini's theorem givesThe odd imaginary part integrates to zero, whileTherefore the expression equalswith the quotient interpreted continuously at $X=0$.

For $|y|>2$, one has $2(1-\sin y/y)\ge 1$. Applying this with $y=uX$ and using part (i),

Continuity of $\phi$ at zero and $\phi (0)=1$ allow $u>0$ so small thatPointwise convergence and dominated convergence make the corresponding expression for ${\phi }_n$ less than $\epsilon$ for all sufficiently large $n$. Part (ii) then boundsThere are only finitely many remaining $n$, so enlarge $M\ge 2/u$ until each of their tails is also at most $\epsilon$. This proves uniform tightness.