Basu's theorem in statistics.  You have a parametrized family of probability distributions.  Call the parameter $\theta$.  The probability distribution of a random variable $X$ depends on $\theta$.

A thing like $X-\theta$ is a random variable but is not a "statistic", since its value---not just its probability distribution---depends on $\theta$.  Think $\theta$ as some unobservable quantity like the average height of 21-year-old males and $X$ as the observable $n$-tuple of heights of 21-year-old males in a random sample.  The average of the heights in the sample would be an _observable_ random variable, i.e. it depends on the pair $(X,\theta)$ only through $X$.

Now a _statistic_ $g(X)$ (i.e. an observable random variable) whose probability distribution does not depend on $\theta$ is called an _ancillary statistic_.

A statistic $g(X)$ that "admits no unbiased estimator of zero" is called a _complete statistic_.  The phrase "admits no unbiased estimator of zero" means there is no function (not depending on $\theta$) $h$ such that the expected value of $h(g(X))$ is $0$ regardless of the value of $\theta$.

A statistic $g(X)$ is called a _sufficient statistic_ if the conditional probability distribution of $X$ given $g(X)$ does not depend on $\theta$.  (Intuitively, $g(X)$ contains all information available in $X$ that is relevant to drawing inferences about the value of $\theta$.

Basu's theorem says every complete sufficient statistic is independent of every ancillary statistic.

I have seen at least one instance in which the only convenient way to prove two particular random variables are independent is to embed their probability distribution into a parametrized family of probability distributions in such a way that one of them becomes a complete sufficient statistic and the other becomes an ancillary statistic.

I'm suddenly realizing that I don't entirely remember the details of that example.  To be continued.....