Non-uniform Berry Essen Bounds

Twice in the last week, I’ve had computer scientists laugh about the insanity of asymptotics in Statistics¹. The insanity comes from the assumption that is never true: \(n = \infty\). As someone working on asymptotic approximations in Statistics, I feel the need to defend its usefulness by showing what people have done in the past, but also think through ways to improve upon the justification for using normal models.

\(\Delta\) isn’t Zero

While it appears computer scientists don’t like when \(n = \infty\), they seem to be ok letting \(\Delta = 0\) when analyzing a complex non-linear system. If you have some complex function \(f(x)\), let’s just analyze the second order expansion. PID controllers are second order controllers and are by far the most popular type of controller.

It is interesting that scientists are more ok allowing \(\Delta = 0\) instead of letting \(n = \infty\)². My gut is happier with the former situation. The funny thing is they are the same. Assuming \(\Delta = 0\) is the same as assuming \(n = \infty\). The proof I learned of the CLT³ is the Lindeberg swapping trick. You take two derivatives (hence the need for two finite moments) and write all higher order terms as \(o(1)\).

Footnotes

1. I remark that computer scientists are apparently very against normal approximations and enjoy their finite sample correct concentration inequalities. This tracks with my earlier post on asymptotics.

2. Perhaps it is because humans interact with \(0\) but never with \(\infty\)

3. And I think a very nice one that gets at the intuition of what the CLT does.