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 powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials.

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

Definition

Assume that X₁, ..., X_k are random variables with finite moments. The Wick product

[math]\displaystyle{ \langle X_1,\dots,X_k \rangle\, }[/math]

is a sort of product defined recursively as follows:

[math]\displaystyle{ \langle \rangle = 1\, }[/math]

(i.e. the empty product—the product of no random variables at all—is 1). For k ≥ 1, we impose the requirement

[math]\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, }[/math]

where [math]\displaystyle{ \widehat{X}_i }[/math] means that X_i is absent, together with the constraint that the average is zero,

[math]\displaystyle{ \operatorname{E} \langle X_1,\dots,X_k\rangle = 0. \, }[/math]

Equivalently, the Wick product can be defined by writing the monomial [math]\displaystyle{ X_1\dots X_k }[/math] as a "Wick polynomial":

[math]\displaystyle{ X_1\dots X_k = \sum_{S\subseteq\left\{1,\dots,k\right\}} \operatorname{E}\left(\textstyle\prod_{i\notin S} X_i\right) \cdot \langle X_i : i \in S \rangle \, }[/math],

where [math]\displaystyle{ \langle X_i : i \in S \rangle }[/math] denotes the Wick product [math]\displaystyle{ \langle X_{i_1},\dots,X_{i_m} \rangle }[/math] if [math]\displaystyle{ S = \left\{i_1,\dots,i_m\right\} }[/math]. This is easily seen to satisfy the inductive definition.

Examples

It follows that

[math]\displaystyle{ \langle X \rangle = X - \operatorname{E}X,\, }[/math]

[math]\displaystyle{ \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),\, }[/math]

[math]\displaystyle{ \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\\ &-\operatorname{E}(XYZ)\\ &+2\operatorname{E}(XY)\operatorname{E}Z+2\operatorname{E}(XZ)\operatorname{E}Y+2\operatorname{E}(YZ)\operatorname{E}X\\ &-6(\operatorname{E}X)(\operatorname{E}Y)(\operatorname{E}Z). \end{align} }[/math]

Another notational convention

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

[math]\displaystyle{ : X_1, \dots, X_k:\, }[/math]

and the angle-bracket notation

[math]\displaystyle{ \langle X \rangle\, }[/math]

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

[math]\displaystyle{ X'^n = \langle X,\dots,X \rangle\, }[/math]

with n factors.

The sequence of polynomials P_n such that

[math]\displaystyle{ P_n(X) = \langle X,\dots,X \rangle = X'^n\, }[/math]

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

[math]\displaystyle{ P_n'(x) = nP_{n-1}(x),\, }[/math]

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

[math]\displaystyle{ X'^n = B_n(X)\, }[/math]

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

[math]\displaystyle{ X'^n = H_n(X)\, }[/math]

where H_n is the nth Hermite polynomial.

Binomial theorem

[math]\displaystyle{ (aX+bY)^{'n} = \sum_{i=0}^n {n\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}} }[/math]

Wick exponential

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

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.