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 X₁, ..., X_k are random variables with finite moments. The Wick product

$${\displaystyle \langle X_{1},\dots ,X_{k}\rangle \,}$$

is a sort of product defined recursively as follows:^{[ citation needed ]}

$${\displaystyle \langle \rangle =1\,}$$

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

$${\displaystyle {\partial \langle X_{1},\dots ,X_{k}\rangle \over \partial X_{i}}=\langle X_{1},\dots ,X_{i-1},{\widehat {X}}_{i},X_{i+1},\dots ,X_{k}\rangle ,}$$

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

$${\displaystyle \operatorname {E} {\bigl [}\langle X_{1},\dots ,X_{k}\rangle {\bigr ]}=0.\,}$$

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

$${\displaystyle X_{1}\dots X_{k}=\!\!\sum _{S\subseteq \left\{1,\dots ,k\right\}}\!\!\operatorname {E} \left[\textstyle \prod _{i\notin S}X_{i}\right]\cdot \langle X_{i}:i\in S\rangle ,}$$

where ${\displaystyle \langle X_{i}:i\in S\rangle }$ denotes the Wick product ${\displaystyle \langle X_{i_{1}},\dots ,X_{i_{m}}\rangle }$ if ${\displaystyle S=\left\{i_{1},\dots ,i_{m}\right\}.}$ This is easily seen to satisfy the inductive definition.

Examples

It follows that

$${\displaystyle {\begin{aligned}\langle X\rangle =&\ X-\operatorname {E} [X],\\[4pt]\langle X,Y\rangle =&\ XY-\operatorname {E} [Y]\cdot X-\operatorname {E} [X]\cdot Y+2(\operatorname {E} [X])(\operatorname {E} [Y])-\operatorname {E} [XY],\\[4pt]\langle X,Y,Z\rangle =&\ XYZ\\&-\operatorname {E} [Y]\cdot XZ\\&-\operatorname {E} [Z]\cdot XY\\&-\operatorname {E} [X]\cdot YZ\\&+2(\operatorname {E} [Y])(\operatorname {E} [Z])\cdot X\\&+2(\operatorname {E} [X])(\operatorname {E} [Z])\cdot Y\\&+2(\operatorname {E} [X])(\operatorname {E} [Y])\cdot Z\\&-\operatorname {E} [XZ]\cdot Y\\&-\operatorname {E} [XY]\cdot Z\\&-\operatorname {E} [YZ]\cdot X\\&-\operatorname {E} [XYZ]\\&+2\operatorname {E} [XY]\operatorname {E} [Z]\\&+2\operatorname {E} [XZ]\operatorname {E} [Y]\\&+2\operatorname {E} [YZ]\operatorname {E} [X]\\&-6(\operatorname {E} [X])(\operatorname {E} [Y])(\operatorname {E} [Z]).\end{aligned}}}$$

Another notational convention

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

$${\displaystyle :X_{1},\dots ,X_{k}:\,}$$

and the angle-bracket notation

$${\displaystyle \langle X\rangle \,}$$

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

$${\displaystyle X'^{n}=\langle X,\dots ,X\rangle \,}$$

with n factors.

The sequence of polynomials P_n such that

$${\displaystyle P_{n}(X)=\langle X,\dots ,X\rangle =X'^{n}\,}$$

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

$${\displaystyle P_{n}'(x)=nP_{n-1}(x),\,}$$

for n = 0, 1, 2, ... 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

$${\displaystyle X'^{n}=B_{n}(X)\,}$$

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

$${\displaystyle X'^{n}=H_{n}(X)\,}$$

where H_n is the nth Hermite polynomial.

Binomial theorem

$${\displaystyle (aX+bY)^{'n}=\sum _{i=0}^{n}{n \choose i}a^{i}b^{n-i}X^{'i}Y^{'{n-i}}}$$

Wick exponential

$${\displaystyle \langle \operatorname {exp} (aX)\rangle \ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=0}^{\infty }{\frac {a^{i}}{i!}}X^{'i}}$$

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.