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,

$\langle X_1,\dots,X_k \rangle\,$

is a sort of product of the random variables, X₁, ..., X_k, defined recursively as follows:

$\langle \rangle = 1\,$

(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,

${\partial\langle X_1,\dots,X_k\rangle \over \partial X_i} = \langle X_1,\dots,X_{i-1}, \widehat{X}_i, X_{i+1},\dots,X_k \rangle,$

where $\widehat{X}_i$ means X_i is absent, and the constraint that

$\operatorname{E} \langle X_1,\dots,X_k\rangle = 0\mbox{ for }k \ge 1. \,$

Examples

It follows that

$\langle X \rangle = X - \operatorname{E}X,\,$

$\langle X, Y \rangle = X Y - \operatorname{E}Y\cdot X - \operatorname{E}X\cdot Y+ 2(\operatorname{E}X)(\operatorname{E}Y) - \operatorname{E}(X Y).\,$

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

Another notational convention

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

$: X_1, \dots, X_k:\,$

and the angle-bracket notation

$\langle X \rangle\,$

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

$X'^n = \langle X,\dots,X \rangle\,$

with n factors.

The sequence of polynomials P_n such that

$P_n(X) = \langle X,\dots,X \rangle = X'^n\,$

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

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

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

$X'^n = B_n(X)\,$

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

$X'^n = H_n(X)\,$

where H_n is the nth Hermite polynomial.

Binomial theorem

$(aX+bY)^{'n} = \sum_{i=0}^n {n\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}}$

Wick exponential

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

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