Editing Collatz conjecture (section)

===Stopping times===
As proven by [[Riho Terras (mathematician)|Riho Terras]], almost every positive integer has a finite stopping time.<ref name="Terras (1976)"/> In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of [[#As a parity sequence|parity vectors]] and uses the [[central limit theorem]].

In 2019, [[Terence Tao]] improved this result by showing, using logarithmic [[Probability density function|density]], that [[almost all]] (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, ''[[Quanta Magazine]]'' wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".<ref name="Tao"/><ref>{{Cite web |last=Hartnett |first=Kevin |date=December 11, 2019 |title=Mathematician Proves Huge Result on 'Dangerous' Problem |url=https://www.quantamagazine.org/mathematician-proves-huge-result-on-dangerous-problem-20191211/ |website=Quanta Magazine}}</ref>