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n-point correlations of Gaussian random variables can be simplified with Wick expansion. $$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots x_{i_{2n}} e^{- \frac{1}{2}\sum x_i^2} = \sum_{\sigma \in \text{matchings}} \; \prod_{\{i,j\}\in \sigma} \langle x_i x_j \rangle$$ A matching is a partition of $\{ 1,2,\dots ,2n\}$ into two-element sets, e.g. $\{ \{1,2\},\dots, \{2n-1,2n\}\}$.

The Chebyshev polynomials are orthogonal with respect to $\mu(dx) = \sqrt{4-x^2} dx$ but there probably isn't a way to simplify n-point correlations, \[ \int_{S^n} x_{i_1}\dots x_{i_{2n}} \sqrt{1 - (x_1^2+\dots + x_{n}^2)} \; dx \]

Is there an analogue of Wick's formula for other orthogonal polynomials? It seems rather unlikely since the measure can be arbitrary.

Maybe look at Wick expansion as the sum over matchings. Is there a version of the Wick expansion for non-crossing matchings or for Schroder paths? alt text

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1 Answer 1

The integral you want to compute is related to multivariable beta functions, the Dirichlet distribution and Dirichlet integrals. In particular one has $$\int\int\cdots\int f(x_1+\cdots+x_n) x_1^{k_1}\cdots x_n^{k_n} dx_1\cdots dx_n$$ $$=\frac{\Gamma(k_1+1)\cdots \Gamma(k_n+1)}{\Gamma(k_1+\cdots+k_n+n)}\int_0^1 f(t)t^{k_1+\cdots+k_n+n-1}dt$$ where the first integral is over the simplex $\sum x_i\le 1$, and in your case $f(t)=\sqrt{1-t}$. Since you are considering moments the $k_i$'s will end up being half-integers and therefore the result will be some sort of generalized Catalan number.

There is also the "Free probability" point of view, which I think is closer to what you want. In fact the joint distribution of free Gaussian random variables has an expansion similar to the Wick expansion but over noncrossing matchings. Of course in free probability the role of the normal distribution is played by the Wigner semicircle distribution.

From the perspective of orthogonal polynomials I believe the correspondence goes like this: Hermite polynomials correspond to the normal distribution and complete matchings. Chebyshev polynomials of second kind correspond to the semicircle distribution and to noncrossing matchings. Charlier polynomials correspond to set partitions, Laguerre polynomials correspond to permutations etc.

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Any seminal references where the correspondence is explained in some detail? Thanks. –  Andrey Rekalo Sep 12 '11 at 8:18
@Andrey: I've gathered these in bits and pieces. For Hermite polynomials one has the usual Wick expansion as in the OP, for Chebyshev the sum is over non-crossing partitions and this is one of the basic facts in free probability. For Laguerre polynomials (Wishart distribution) one has an expansion as a sum over all permutations of product $\langle x_{i_1},\dots,x_{i_k}\rangle\cdots\langle x_{i_r},\dots,x_{i_n}\rangle$ where $(i_1,\dots,i_k)\dots (i_r,\dots,i_n)$ is the cycle decomposition of the permutation. These formulas are used often in random matrix contexts for example... –  Gjergji Zaimi Sep 12 '11 at 8:29
But I wonder if there is a general Wick expansion that generalizes all these. It can probably be expressed in terms of weighted Motzkin/Schroeder paths though I've never seen it. I'll see if I can derive something by tomorrow. :P –  Gjergji Zaimi Sep 12 '11 at 8:31
@Gjergji Zaimi: Thanks a lot for the comments. –  Andrey Rekalo Sep 12 '11 at 8:42

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