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

$⟨X_1,\dots ,X_k⟩$

is a sort of product defined recursively as follows:

$⟨⟩=1$

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

$\frac{\partial ⟨X_1,\dots ,X_k⟩}{\partial X_i}=⟨X_1,\dots ,X_{i-1},{\hat{X}}_i,X_{i+1},\dots ,X_k⟩,$

where ${\hat{X}}_i$ means that X_i is absent, together with the constraint that the average is zero,

$\mathrm{E}⟨X_1,\dots ,X_k⟩=0.$

Equivalently, the Wick product can be defined by writing the monomial $X_1\dots X_k$ as a "Wick polynomial":

$X_1\dots X_k=\sum _{S\subseteq \{1,\dots ,k\}}\mathrm{E}\left(\prod _{i\notin S}{X}_i\right)\cdot ⟨X_i:i\in S⟩$,

where $⟨X_i:i\in S⟩$ denotes the Wick product $⟨X_{i_1},\dots ,X_{i_m}⟩$ if $S=\{i_1,\dots ,i_m\}$. This is easily seen to satisfy the inductive definition.

Examples

It follows that

$⟨X⟩=X-\mathrm{E}X,$

$⟨X,Y⟩=XY-\mathrm{E}Y\cdot X-\mathrm{E}X\cdot Y+2(\mathrm{E}X)(\mathrm{E}Y)-\mathrm{E}(XY),$

$\begin{array}{rl}⟨X,Y,Z⟩=& XYZ\\ & -\mathrm{E}Y\cdot XZ\\ & -\mathrm{E}Z\cdot XY\\ & -\mathrm{E}X\cdot YZ\\ & +2(\mathrm{E}Y)(\mathrm{E}Z)\cdot X\\ & +2(\mathrm{E}X)(\mathrm{E}Z)\cdot Y\\ & +2(\mathrm{E}X)(\mathrm{E}Y)\cdot Z\\ & -\mathrm{E}(XZ)\cdot Y\\ & -\mathrm{E}(XY)\cdot Z\\ & -\mathrm{E}(YZ)\cdot X\\ & -\mathrm{E}(XYZ)\\ & +2\mathrm{E}(XY)\mathrm{E}Z+2\mathrm{E}(XZ)\mathrm{E}Y+2\mathrm{E}(YZ)\mathrm{E}X\\ & -6(\mathrm{E}X)(\mathrm{E}Y)(\mathrm{E}Z).\end{array}$

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

$⟨X⟩$

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

$X{\text{'}}^n=⟨X,\dots ,X⟩$

with n factors.

The sequence of polynomials P_n such that

$P_n(X)=⟨X,\dots ,X⟩=X{\text{'}}^n$

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

${P_n}^{\prime }(x)=n{P}_{n-1}(x),$

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

$X{\text{'}}^n=B_n(X)$

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

$X{\text{'}}^n=H_n(X)$

where H_n is the nth Hermite polynomial.

Binomial theorem

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

Wick exponential

$⟨\exp (aX)⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _{i=0}^{\mathrm{\infty }}\frac{a^i}{i!}{X}^{\text{'}i}$

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.