Lagmental Vicfred

Characteristic Functions Turn Sums into Products by Vicfred

tags: mathematics, limit theorems, Borel--Cantelli, central limit theorem

The Fourier transform of a distribution always exists and factorizes for independent sums. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Statement

For events \(A_n\), the notation \(A_n\ \mathrm{i.o.}\) means infinitely many occur. For sums \(S_n=X_1+\cdots+X_n\), different normalizations lead to laws of large numbers or central limits.

$$ \phi_X(t)=\mathbf E[e^{itX}] $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ X\perp Y\Longrightarrow\phi_{X+Y}(t)=\phi_X(t)\phi_Y(t) $$

Worked algebra

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \phi_{(S_n-n\mu)/(\sigma\sqrt n)}(t)=\left[e^{-it\mu/(\sigma\sqrt n)}\phi_X\!\left(\frac{t}{\sigma\sqrt n}\right)\right]^n\longrightarrow e^{-t^2/2} $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \phi_X(t)=\mathbf E[e^{itX}],\\[5pt] \mathsf{C}\;&:\quad X\perp Y\Longrightarrow\phi_{X+Y}(t)=\phi_X(t)\phi_Y(t). \end{aligned} $$

Caveat

Convergence almost surely, in probability, in distribution, and in \(L^p\) are distinct. One implication cannot be reversed without extra hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X\perp Y\Longrightarrow\phi_{X+Y}(t)=\phi_X(t)\phi_Y(t) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Tue 07 November 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.