# Wick product

In probability theory, the Wick product is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products.

The definition of the Wick product immediately leads to the Wick power of a single random variable and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power-series expansions by the Wick powers.

The Wick product is named after physicist Gian-Carlo Wick, cf. Wick's theorem.

## Definition

The Wick product,

${\displaystyle \langle X_{1},\dots ,X_{k}\rangle \,}$

is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:[citation needed]

${\displaystyle \langle \rangle =1\,}$

(i.e. the empty product—the product of no random variables at all—is 1). Thereafter finite moments must be assumed. Next, for k≥1,

${\displaystyle {\partial \langle X_{1},\dots ,X_{k}\rangle \over \partial X_{i}}=\langle X_{1},\dots ,X_{i-1},{\widehat {X}}_{i},X_{i+1},\dots ,X_{k}\rangle ,}$

where ${\displaystyle {\widehat {X}}_{i}}$ means Xi is absent, and the constraint that

${\displaystyle \operatorname {E} \langle X_{1},\dots ,X_{k}\rangle =0{\mbox{ for }}k\geq 1.\,}$

## Examples

It follows that

${\displaystyle \langle X\rangle =X-\operatorname {E} X,\,}$
${\displaystyle \langle X,Y\rangle =XY-\operatorname {E} Y\cdot X-\operatorname {E} X\cdot Y+2(\operatorname {E} X)(\operatorname {E} Y)-\operatorname {E} (XY).\,}$
{\displaystyle {\begin{aligned}\langle X,Y,Z\rangle =&XYZ\\&-\operatorname {E} Y\cdot XZ\\&-\operatorname {E} Z\cdot XY\\&-\operatorname {E} X\cdot YZ\\&+2(\operatorname {E} Y)(\operatorname {E} Z)\cdot X\\&+2(\operatorname {E} X)(\operatorname {E} Z)\cdot Y\\&+2(\operatorname {E} X)(\operatorname {E} Y)\cdot Z\\&-\operatorname {E} (XZ)\cdot Y\\&-\operatorname {E} (XY)\cdot Z\\&-\operatorname {E} (YZ)\cdot X\\&-\operatorname {E} (XYZ)\,.\\\end{aligned}}}

## Another notational convention

In the notation conventional among physicists, the Wick product is often denoted thus:

${\displaystyle :X_{1},\dots ,X_{k}:\,}$

and the angle-bracket notation

${\displaystyle \langle X\rangle \,}$

is used to denote the expected value of the random variable X.

## Wick powers

The nth Wick power of a random variable X is the Wick product

${\displaystyle X'^{n}=\langle X,\dots ,X\rangle \,}$

with n factors.

The sequence of polynomials Pn such that

${\displaystyle P_{n}(X)=\langle X,\dots ,X\rangle =X'^{n}\,}$

form an Appell sequence, i.e. they satisfy the identity

${\displaystyle P_{n}'(x)=nP_{n-1}(x),\,}$

for n = 0, 1, 2, ... and P0(x) is a nonzero constant.

For example, it can be shown that if X is uniformly distributed on the interval [0, 1], then

${\displaystyle X'^{n}=B_{n}(X)\,}$

where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then

${\displaystyle X'^{n}=H_{n}(X)\,}$

where Hn is the nth Hermite polynomial.

## Binomial theorem

${\displaystyle (aX+bY)^{'n}=\sum _{i=0}^{n}{n \choose i}a^{i}b^{n-i}X^{'i}Y^{'{n-i}}}$

## Wick exponential

${\displaystyle \langle \operatorname {exp} (aX)\rangle \ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=0}^{\infty }{\frac {a^{i}}{i!}}X^{'i}}$

## References

• Wick Product Springer Encyclopedia of Mathematics
• Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages 767—775, 1987.
• Hida, T. and Ikeda, N. (1967) "Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral". Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press
• Wick, G. C. (1950) "The evaluation of the collision matrix". Physical Rev. 80 (2), 268–272.
• Hu, Yao-zhong; Yan, Jia-an (2009) "Wick calculus for nonlinear Gaussian functionals", Acta Mathematicae Applicatae Sinica (English Series), 25 (3), 399–414 doi:10.1007/s10255-008-8808-0