Establish an invariance principle first

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Establish an invariance principle first

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[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]

(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 $S_n := \frac{X_1+\ldots+X_n}{\sqrt{n}}$ 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 $X_i$.

Indeed, once one has the invariance principle, one simply replaces the $X_i$ with a gaussian iid sequence and uses the base case.

[GENERAL DISCUSSION]

The replacement trick has also been used in random matrix theory; see [http://terrytao.wordpress.com/2008/08/02/random-matrices-universality-of-esds-and-the-circular-law/ this blog post] for some further discussion.

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