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. I think it was something quite similar to Lemma 3 on page 7 of this: http://www.stat.duke.edu/~sd83/Research/das-dey-mult-gamma.pdf