Stochastic Processes [STOKD] Closed

There is an interesting link between probability distributions and orthogonal polynomials. For instance, consider the Hermite polynomials, H0,H1,…. These are orthogonal with respect to the weight function e−x2/2 on the support (−∞,∞). In other words, ∫∞−∞Hi(x)Hj(x)e−x2/2dx=2π−−√i!δij.

The weighting function and the coefficient having a factor 2π−−√ should ring a bell; specifically, you find these represented in a standard normal distribution.

Norbert Wiener recognized that the two are intricately linked and developed what is called the Gauss-Hermite Polynomial Chaos. In short, we can write a random variable k of any distribution as an infinite series about a random variable ζ of a normal (Gaussian) distribution by using the Hermite polynomials as basis functions:
k=∑i=0∞kiHi(ζ).
It turns out, you can generalize this to other distributions, and to discrete distributions.