Wick product explained

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

$$\backslash langle\; X\_1,\backslash dots,X\_k\; \backslash rangle\backslash ,$$

is a sort of product defined recursively as follows:

$$\backslash langle\; \backslash rangle\; =\; 1\backslash ,$$

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

$$=\; \backslash langle\; X\_1,\backslash dots,X\_,\; \backslash widehat\_i,\; X\_,\backslash dots,X\_k\; \backslash rangle,$$

where

\widehat{X} _{i} means that is absent, together with the constraint that the average is zero,

$$\backslash operatorname\; \backslash bigl[\backslash langle\; X\_1,\backslash dots,X\_k\backslash rangle\; \backslash bigr]\; =\; 0.\; \backslash ,$$

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

$$X\_1\backslash dots\; X\_k\; =\; \backslash !\backslash !\; \backslash sum\_\; \backslash !\backslash !\; \backslash operatorname\backslash left[\backslash textstyle\backslash prod\_\{i\backslash notin\; S\}\; X\_i\backslash right]\; \backslash cdot\; \backslash langle\; X\_i\; :\; i\; \backslash in\; S\; \backslash rangle,$$

where

\langle X _{i} : i \in S \rangle denotes the Wick product \langle X _{i _{1}} , ... ,X _{i _{m}} \rangle if S = \left\{i _{1,...,i} _{m\right\}.} This is easily seen to satisfy the inductive definition.

Examples

It follows that

Another notational convention

In the notation conventional among physicists, the Wick product is often denoted thus:

$$:\; X\_1,\; \backslash dots,\; X\_k:\backslash ,$$

and the angle-bracket notation

$$\backslash langle\; X\; \backslash rangle\backslash ,$$

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

$$X\text{'}^n\; =\; \backslash langle\; X,\backslash dots,X\; \backslash rangle\backslash ,$$

with factors.

The sequence of polynomials such that

$$P\_n(X)\; =\; \backslash langle\; X,\backslash dots,X\; \backslash rangle\; =\; X\text{'}^n\backslash ,$$

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

$$P\_n\text{'}(x)\; =\; nP\_(x),\backslash ,$$

for and is a nonzero constant.

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

$$X\text{'}^n\; =\; B\_n(X)\backslash ,$$

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

$$X\text{'}^n\; =\; H\_n(X)\backslash ,$$

where is the th Hermite polynomial.

Binomial theorem

$$(aX+bY)^\; =\; \backslash sum\_^n\; a^ib^\; X^\; Y^$$

Wick exponential

$$\backslash langle\; \backslash operatorname(aX)\backslash rangle\; \backslash \; \backslash stackrel\; \backslash \; \backslash sum\_^\backslash infty\backslash frac\; X^$$

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.