Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 27G / b

The convergence $X_n\to X$ in distribution meansfor every continuity point $x$ of $F_X$.

Let $F^{-1}(u)=\inf \{x:u\le F(x)\}$. The generalized-inverse identitygivesand similarly $\mathbb{P}({\tilde{X}}_n\le x)=F_{X_n}(x)$. Thus the coupled variables have the required marginal distributions.

At every continuity point $u$ of the nondecreasing function $F_X^{-1}$, convergence of the distribution functions at continuity points of $F_X$ impliesIndeed, values just below and above $F_X^{-1}(u)$ can be chosen at continuity points of $F_X$ and trap the two generalized inverses. A monotone function has at most countably many discontinuities, so this convergence holds for Lebesgue-almost every $u\in (0,1)$. Therefore ${\tilde{X}}_n\to \tilde{X}$ almost surely. This is the quantile coupling for convergence in distribution.