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 X₁, ..., 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

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

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

$\operatorname {E} \langle X_{1},\dots ,X_{k}\rangle =0.\,$

Equivalently, the Wick product can be defined by writing the monomial $X_{1}\dots X_{k}$ 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

$\langle X\rangle =X-\operatorname {E} X,\,$

$\langle X,Y\rangle =XY-\operatorname {E} Y\cdot X-\operatorname {E} X\cdot Y+2(\operatorname {E} X)(\operatorname {E} Y)-\operatorname {E} (XY),\,$

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

Another notational convention

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

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

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 P_n such that

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

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

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

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

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

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

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

where H_n is the nth Hermite polynomial.

Binomial theorem

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

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.