Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 28K / c / Solution

Apply part (b) simultaneously at every rational $t$; the intersection of these probability-one events still has probability one. On that event, monotonicity and continuity extend the convergence from the rationals to every real $t$.

More directly, fix $\epsilon >0$. Since $F$ is a continuous cumulative distribution function, choose rational pointsso that the two tails have $F(t_0)<\epsilon$ and $1-F(t_k)<\epsilon$, while every increment $F(t_i)-F(t_{i-1})<\epsilon$. Part (b) makes $|F_m^{\ast }(t_i)-F(t_i)|<\epsilon$ at all these finitely many points for all sufficiently large $m$. If $t_{i-1}\le t\le t_i$, monotonicity givesso $|F_m^{\ast }(t)-F(t)|<2\epsilon$; the same estimate follows in the tails. This proves the uniform convergence of distribution functions to a continuous limit:almost surely.

Solved by gpt-5.6-sol high.