Limit of CDFs and inverse CDFs for transformed random variables

Question

Assume that $X$ is continuous random variable with CDF $F$. Let $t$ and $t_n$ be two functions, so that the transformed random variables $t(X)$ and $t_n(X)$ are well-defined for all $n$, and that $t_n \to t$ when $n \to \infty$.

Is is true that $F_{t_n}(x) \to F_t (x)$ and that $F^{-1}_{t_n}(x) \to F^{-1}_t(x)$ when $n \to \infty$?

What are the assumptions that are needed for $t_n$ and $t$? Does it complicate matters if $t_n$ is discontinuous?

I was thinking applying (reverse) Fatou's lemma for $F_{t_n}(x) = E_x[1(t_n(X) < x)]$ but I'm unsure if there is a better argument and if this even works for the inverse CDFs.

score 2 · Accepted Answer · answered Oct 25 '21 at 10:14

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If $t_n\to t$ pointwise, for any bounded continuous function $f$, \begin{align} \lim_{n\to\infty}\mathsf{E}f(t_n(X))&=\mathsf{E}\lim_{n\to\infty}f(t_n(X)) \\ &=\mathsf{E}f\left(\lim_{n\to\infty} t_n(X)\right) \\ &=\mathsf{E}f(t(X)) \end{align} by the dominated convergence theorem, which implies that $t_n(X)\xrightarrow{d}t(X)$. As for the convergence of quantiles, see, e.g., this question.

answered Oct 25 '21 at 10:14

What's the dominating function which you use for the dominated convergence theorem? – Idontgetit Oct 25 '21 at 10:22
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@Idontgetit $f$ is bounded... – Oct 25 '21 at 10:30
Does this hold if the convergence $t_n \to t$ is in probability? – NPHA Oct 25 '21 at 10:47
@NPHA Yes. https://math.stackexchange.com/questions/3374830/dominated-convergence-theorem-with-almost-surely-replaced-by-convergence-in-p – Oct 25 '21 at 10:50

Limit of CDFs and inverse CDFs for transformed random variables

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