Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 26L (Probability and Measure)

$X_n\Rightarrow X$ means $Ef(X_n)\to Ef(X)$ for every bounded continuous $f$, equivalently convergence of distribution functions at continuity points of $X$.

Characteristic-function convergence theorem says $X_n\Rightarrow X$ iff ${\Phi }_{X_n}(t)\to {\Phi }_X(t)$ for all $t$, where the limiting function is continuous at zero.

Absolute convergence gives $S_n\to S$ almost surely. Put $B_j=(X_j+1)/2$. Then $S=2{\sum }_j{B}_j/2^j-1$. Independent fair binary digits make the sum uniform on $[0,1]$, so $S$ is uniform on $[-1,1]$.

Again the series converges absolutely. $T=2{\sum }_j{B}_j/3^j$ has the Cantor distribution. Every particular digit sequence has probability zero, so it has no atoms; its support is the Cantor set, which has Lebesgue measure zero, so it cannot have a Lebesgue density.