Limiting distribution of $G_n(X_n)$

Question

Consider two sequences of random variables. At each point in the sequence $X_n \sim F_n$ and $Y_n \sim G_n$, and let $F_n(t)$ and $G_n(t)$ denote their respect CDFs. The distributions $(F_n, G_n)$ are not guaranteed to be continous.

Suppose that both random variables converge to the same limiting distribution. Formally, as $n \to \infty$, $X_n \to^d Z \sim H$ and $Y_n \to^d Z \sim H$, where $H(t)$ is a strictly increasing, continuous function.

Can we show that $G_n(X_n)$ converges to a uniform? If not, can we find a counter-example?

Answer 1

Since $G_n(r) \leqslant 1$ for all $r \in \mathbb{R}$, this function is "dominated" by the unit function over its entire domain. This allows us to apply the dominated convergence theorem to move the limit inside the probability operator. Taking any value $0 \leqslant t \leqslant 1$ we then have:

$$\begin{align} \lim_{n \rightarrow \infty} \mathbb{P} [ G_n(X_n) \leqslant t ] &= \mathbb{P} \bigg[ \lim_{n \rightarrow \infty} G_n(X_n) \leqslant t \bigg] \\[6pt] &= \mathbb{P} \bigg[ H(Z) \leqslant t \bigg] \\[6pt] &= t, \\[6pt] \end{align}$$

which proves pointwise convergence to the uniform.