Establish an invariance principle first

Quick description

To prove a property $P(x)$ for all $x$ , first show that $P(x)$ is equivalent to $P(y)$ for all $x, y$ in the desired parameter space. Then, one only needs to verify $P(x)$ for a single $x$ , which one can choose to make the verification as easy as possible.

Prerequisites

Probability theory

Example 1

(Lindeberg replacement trick) Suppose one wants to prove the central limit theorem, viz. that if $X_1,X_2,\ldots$ is a sequence of iid real-valued random variables with mean zero and variance $1$ , then the random variables $= \frac{X_1+\ldots+X_n}{\sqrt{n}}$ converge in distribution to the standard Gaussian random variable $N(0,1)$ . For simplicity let us assume that all moments of the $X_i$ are finite. Lindeberg's proof of the central limit theorem proceeds in two steps:

(Base case) Verify the central limit theorem in the special case when the $X_i$ are iid gaussians, $X_i \equiv N(0,1)$ . In this case the theorem is easy, basically because the sum of two independent gaussians is still a gaussian.
(Invariance) Show that for each $k$ , the asymptotic limit $\lim_{n \to \infty} {\Bbb E} S_n^k$ of the $k^{th}$ moments of $S_n$ remain unchanged if one replaces the iid sequence $X_i$ by any other iid sequence, say $Y_i$ , with the same mean and variance. This is done by expanding out ${\Bbb E} S_n^k$ and observing that most terms only involve at most two factors of each $X_i$ , and so (by the iid hypothesis) only involve the first and second moments of the $X_i$ .