bibkey: durrett2019probability
authors: Rick Durrett
year: 2019
title: "Probability: Theory and Examples, Fifth Edition"
doi: 10.1017/9781108591034
url: https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf
claim: "Exercise 3.2.9 gives uniform convergence to a continuous distribution function; Theorem 3.2.9 gives bounded-continuous weak tests and Theorem 4.6.3 uses continuous clipping and absolute tails."
strata_touched:
  - D5/S3/Analytic/Probability/UniformCDFConvergence
  - D5/S3/Analytic/Probability/FirstMomentWeakLimit
license: citation-only
triage: anchor

Probability: Theory and Examples

Verified locator

DOI: 10.1017/9781108591034

Source: https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf

Cited result

Exercise 3.2.9, printed page 124:

If $F_n\Rightarrow F$ and $F$ is continuous then ${\sup }_x|F_n(x)-F(x)|\to 0$.

The delivered uniform-CDF theorem states the monotonicity, boundedness and tail conditions explicitly. Its proof uses a finite compact cover and the two tails. The characteristic-function corollary combines this result with Mathlib’s Lévy continuity theorem. The proof of Theorem 2.4.9, printed page 79, also states the pointwise-to-uniform principle for nondecreasing functions.

First-moment truncation

Section 4.6, printed pages 244–246, defines uniform integrability using absolute first-moment tails. The proof of Theorem 4.6.3 on printed page 245 uses the continuous clamp ${\phi }_M(x)=\max (-M,\min (M,x))$ and the inequality $|x-{\phi }_M(x)|\le |x|1_{\{|x|>M\}}$. Theorem 3.2.9 on printed page 119 characterizes weak convergence by bounded continuous real tests. These are the classical ingredients for the varying-law truncation argument; Theorem 4.6.3 itself assumes convergence in probability and is not cited as a verbatim varying-law theorem.

FirstMomentWeakLimit.tendsto_integral_displacement applies this method to Borel probability measures on the real plane. Weak convergence, integrable coordinates for every law and its limit, and uniform absolute coordinate tails imply convergence of the integral of $(y-x)h(x)$ for each bounded continuous real $h$. The bounded continuous approximant is $({\phi }_M(y)-{\phi }_M(x))h(x)$, with the joint law unchanged. Its sequence error is bounded by the sum of the coordinate tails times the bound on $h$; dominated convergence controls its limit-law error. The Lean proof directly uses pinned Mathlib bounded-continuous weak tests, Bochner integral inequalities and dominated convergence. No source code is copied from the book, and no originality claim is made.