Wick product

In probability theory, the Wick product, named for Italian physicist Gian-Carlo Wick, 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 expansion 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.

Definition

Assume that Template:Math are random variables with finite moments. The Wick product

$$⟨X_1,\dots ,X_k⟩$$

is a sort of product defined recursively as follows:Template:Citation needed

$$⟨⟩=1$$

(i.e. the empty product—the product of no random variables at all—is 1). For Template:Math, 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 Template:Mvar 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 Template:Math 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

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

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 Template:Mvar.

Wick powers

The Template:Mvarth Wick power of a random variable Template:Mvar is the Wick product

$$X{\text{'}}^n=⟨X,\dots ,X⟩$$

with Template:Mvar factors.

The sequence of polynomials Template:Mvar 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 Template:Math and Template:Math is a nonzero constant.

For example, it can be shown that if Template:Mvar is uniformly distributed on the interval Template:Math, then

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

where Template:Mvar is the Template:Mvarth-degree Bernoulli polynomial. Similarly, if Template:Mvar is normally distributed with variance 1, then

$$X{\text{'}}^n=H{e}_n(X)$$

where Template:Mvar is the Template:Mvarth probabilist's 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.