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 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 , we impose the requirement

$= \langle X_1,\dots,X_, \widehat_i, X_,\dots,X_k \rangle,$

where $\widehat_i$ means that is absent, together with the constraint that the average is zero,

\operatorname \bigl langle X_1,\dots,X_k\rangle \bigr= 0. \,

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

X_1\dots X_k = \!\! \sum_ \!\! \operatorname\left textstyle\prod_ X_i\right\cdot \langle X_i : i \in S \rangle ,

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

Examples

It follows that

\begin
\langle X \rangle =&\ X - \operatorname \\ pt \langle X, Y \rangle =&\ XY - \operatorname \cdot X - \operatorname \cdot Y + 2(\operatorname (\operatorname - \operatorname Y \\ pt \langle X,Y,Z\rangle =&\ XYZ \\
&- \operatorname \cdot XZ \\
&- \operatorname \cdot XY \\
&- \operatorname \cdot YZ \\
&+ 2(\operatorname (\operatorname \cdot X \\
&+ 2(\operatorname (\operatorname \cdot Y \\
&+ 2(\operatorname (\operatorname \cdot Z \\
&- \operatorname Z\cdot Y \\
&- \operatorname Y\cdot Z \\
&- \operatorname Z\cdot X \\
&- \operatorname YZ\
&+ 2\operatorname Yoperatorname \\
&+ 2\operatorname Zoperatorname \\
&+ 2\operatorname Zoperatorname \\
&- 6(\operatorname (\operatorname (\operatorname .
\end

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 .

Wick powers

The th Wick power of a random variable is the Wick product

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

with factors.

The sequence of polynomials such that

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

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

$P_n'(x) = nP_(x),\,$

for and is a nonzero constant.

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

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

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

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

where is the th Hermite polynomial.

Binomial theorem

$(aX+bY)^ = \sum_^n a^ib^ X^ Y^$

Wick exponential

$\langle \operatorname(aX)\rangle \ \stackrel \ \sum_^\infty\frac X^$

{{No footnotes, date=May 2012

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.

Algebra of random variables