Wick product

Definition

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

\langle X_1,\dots,X_k \rangle,

is a sort of product defined recursively as follows:

\langle \rangle = 1,

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

{\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 that Xi is absent, together with the constraint that the average is zero,

\operatorname{E} \bigl[\langle X_1,\dots,X_k\rangle \bigr] = 0. ,

Equivalently, the Wick product can be defined by writing the monomial X1, ..., Xk as a "Wick polynomial":

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 ,

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

Examples

It follows that

\begin{align} \langle X \rangle =&\ X - \operatorname{E}[X], \[4pt] \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], \[4pt] \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} Perhaps the next several could be added here. --

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

X'^n = He_n(X),

where Hen is the nth probabilist's 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.