CLT as a special case of the Banach fixed-point theorem?

Question

I'm wondering if the Lindeberg–Lévy CLT can be seen as a special case of the Banach fixed-point theorem. I have in mind some map $T:P\rightarrow P$ for some complete metric space of CDFs $P$ with finite variances.

$T$ might be something like: $T(F)(x)=(F\star F)(\sqrt{2}\sigma(F)x+2\mu(F))$, where $\star$ is convolution, $\mu(F):=\int_\mathbb{R} x\ \mathrm{d}F(x)$ is the expectation of $F$ and $\sigma(F)$ is the standard deviation of $F$.

This map is intended to produce the CDF resulting from adding an i.i.d. copy of the original random variable to the original random variable and then normalising so that the resulting random variable has mean 0 and standard deviation 1. By the CLT, $T^n(F)$ should converge to $\mathcal{N}(0,1)$ (so long as $F$ has finite variance).

Intuitively, it seems plausible to me that $T$ is a contraction for some metric on $P$ because the convolution averages things out to make the distribution a bit more bell-shaped and the normalisation prevents repeated contractions from blowing up and on the "same scale''. An obvious choices of metric would be that induced by the $L^2$-norm (since we want finite variance). I haven't checked this though and would appreciate a sanity check on whether this makes sense / seems plausible.

Choosing a metric that makes $P$ complete seems hard. For example, one can easily imagine a sequence of distributions with finite means and variances that converges to a distribution with infinite variance (e.g. a sequence of Pareto distributions with decreasing shape parameters). I'm not sure this is fatal though. The trouble with an incomplete metric space is that a contraction could converge to a point outside the space -- but in this case, we know that the "contraction" converges to a point in the relevant space. So even if this approach fails to be a special case of the Banach fixed-point theorem, it could still be a case of a convergent contraction.

This question is similar and the answer points to Stein's method, which is interesting but different to the method I've suggested. I'm wondering if my method could lead anywhere. I'm asking this purely out of curiosity and fascination with CLTs (and fixed-point theorems!); I don't have any particular application in mind.

Answer 1

Me: I have a question about maths, probability, real analysis. Can the central limit theorem be viewed as a fixed point theorem? In particular, a type of contraction mapping? The mapping is on the space of cumulative distribution functions with finite variance and maps the function F to the convolution of F with itself, but normalised so that the resulting cumulative distribution function is for a random variable with mean 0 and standard deviation 1.

ChatGPT: Yes, the central limit theorem can indeed be viewed as a fixed-point theorem. Specifically, it can be seen as a result of the fixed-point theorem for stochastic processes, which is sometimes referred to as the Doeblin-Fortet-Mourier theorem. This theorem provides conditions under which a sequence of stochastic processes converges to a fixed process in a certain sense.

In the case of the central limit theorem, the sequence of stochastic processes is given by the sum of a large number of independent random variables with the same distribution. The fixed process to which this sequence converges in distribution is the normal distribution. This can be seen as a result of the fact that the convolution operation, which is used in the central limit theorem to combine the distributions of the individual random variables, is a contraction mapping on the space of probability measures with finite variance.

More precisely, if we let F_n denote the distribution function of the sum of n independent random variables with the same distribution, then the central limit theorem states that the sequence of functions {F_n} converges pointwise to the standard normal distribution function as n goes to infinity. This can be viewed as a fixed-point equation of the form F = T(F), where T is the convolution operator normalized to produce a distribution with mean 0 and standard deviation 1.

The fixed-point theorem for stochastic processes provides conditions under which such an equation has a unique solution, which in this case corresponds to the standard normal distribution. These conditions typically involve some form of uniformity or continuity of the operator T, which can be established using techniques from real analysis and probability theory.