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 product is named after physicist Gian-Carlo Wick, cf. Wick's theorem.

Definition

The Wick product,

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

is a sort of product of the random variables, X₁, ..., X_k, defined recursively as follows:

\langle \rangle =1\,

(i.e. the empty product—the product of no random variables at all—is 1). Thereafter finite moments must be assumed. Next, for k≥1,

{\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 X_i is absent, and the constraint that

\operatorname {E}\langle X_{1},\dots ,X_{k}\rangle =0{\mbox{ for }}k\geq 1.\,

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)\,.\\\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.
Hu, Yao-zhong; Yan, Jia-an (2009) "Wick calculus for nonlinear Gaussian functionals", Acta Mathematicae Applicatae Sinica (English Series), 25 (3), 399–414 doi:10.1007/s10255-008-8808-0

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