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 ''X''₁, ..., ''X''_''k'' 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

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

where $\widehat_i$ means that ''X''_''i'' is absent, together with the constraint that the average is zero,

:$\operatorname \langle X_1,\dots,X_k\rangle = 0. \,$

Examples

It follows that

:$\langle X \rangle = X - \operatornameX,\,$

:$\langle X, Y \rangle = X Y - \operatornameY\cdot X - \operatornameX\cdot Y+ 2(\operatornameX)(\operatornameY) - \operatorname(X Y).\,$

:$\begin \langle X,Y,Z\rangle =&XYZ\\ &-\operatornameY\cdot XZ\\ &-\operatornameZ\cdot XY\\ &-\operatornameX\cdot YZ\\ &+2(\operatornameY)(\operatornameZ)\cdot X\\ &+2(\operatornameX)(\operatornameZ)\cdot Y\\ &+2(\operatornameX)(\operatornameY)\cdot Z\\ &-\operatorname(XZ)\cdot Y\\ &-\operatorname(XY)\cdot Z\\ &-\operatorname(YZ)\cdot X\\ &-\operatorname(XYZ)\\ &+2\operatorname(XY)\operatornameZ+2\operatorname(XZ)\operatornameY+2\operatorname(YZ)\operatornameX\\ &-6(\operatornameX)(\operatornameY)(\operatornameZ). \end$

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 ''X''.

Wick powers

The ''n''th 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 ''P''_''n'' 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 ''n'' = 0, 1, 2, ... and ''P''₀(''x'') is a nonzero constant.

For example, it can be shown that if ''X'' is uniformly distributed on the interval , 1 then

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

where ''B''_''n'' is the ''n''th-degree Bernoulli polynomial. Similarly, if ''X'' is normally distributed with variance 1, then

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

where ''H''_''n'' is the ''n''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