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Let $X, Y$ be two random variables with laws $F$ and $G$ respectively. The Kolmogorov distance between these two laws is defined as: $$ d_{Kol}(F, G) = \sup_{x \in \mathbb R} |\mathbb P(X \leq x) - \mathbb P(Y \leq x)| $$ Now consider a sequence of random variables $X_n \sim F_n$ and a random variable $X \sim F$. It is easy to see that $d_{Kol}(F_n, F) \rightarrow 0$ implies $X_n \rightarrow_d X$ ($\rightarrow_d$ is for convergence in distribution). My questions are:

  1. Under what assumptions, the other direction is also true? (that is, convergence in distribution implies convergence in Kolmogorov distance). This paper (the first couple of paragraphs) claims this to be true when $F$ is continuous, but I got lost when trying to prove it when using the second Dini's theorem.
  2. In general, can convergence in probability imply convergence in Kolmogorov distance (without assuming continuous target law)?

Any ideas and insights would be really appreciated.

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  • $\begingroup$ @Glen_b Thanks for pointing it out! I didn't notice it until you brought it up. The statement in 1. has already been corrected. $\endgroup$
    – Ethan
    Jan 30 at 3:13

2 Answers 2

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For 1 with $F_{\infty}$ absolute continuous to the Lebesgue measure, you could check out exercise 3.2.9 in Cyrus Maz's solution to Durrett's Probability. The basic idea is that since $F_{n}$ and $F_{\infty}$ are monotone and their range $[0,1]$ is compact, you can cover $[0,1]$ by finitely many $\epsilon$-balls centered at $F(x_i)$'s for some carefully chosen $x_i$'s, and control the convergence at these points to get a small Komogorov distance. In the end, pointwise convergence + monotonicity of functions + compact ranges implies uniform convergence, I think this is why the paper mentions Dini's Theorem.

2 won't hold without assuming continuous target law. For a counterexample, consider $X_n$ with point mass at $\frac{1}{n}$ and $X_{\infty}$ with point mass at $0$. Then $X_n$ converge to $X_{\infty}$ in probability. But for each $n$, $|\mathbb{P}(X_n \leq 0) - \mathbb{P}(X_{\infty}\leq 0)| = |0 - 1| = 1$ for all $n$, so $d_{Kol}(F_n, F) \geq 1$ for all $n$.

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The distinction between convergence in distribution and convergence in Kolmogorov norm is the reason for the importance of the Glivenko-Cantelli theorem on convergence of the empirical cdf $\mathbb{F}_n$ to the true cdf $F$.

Convergence in distribution implies convergence of $F_n$ to $F$ pointwise at points where $F$ is continuous (and uniformly on intervals that don't contain a discontinuity of $F$). But the Glivenko-Cantelli theorem is stronger, and says the convergence is uniform on the whole domain even in the presence of discontinuities. In contrast to the counterexample with convergence in distribution, where the discontinuity in $F_n$ moves as $n$ increases, the discontinuities in $\mathbb{F}_n$ (the ones that are due to discontinuity of $F$, not just finite sample size) stay in the same place.

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