Wick product explained

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,...,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,...,X} _{k\rangle} \over \partial X _{i} =} \langle X _{1,...,X} _{i-1} , \widehat{X} _{i,} X _{i+1} , ... ,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,...,X} _{k\rangle} = 0 .

Equivalently, the Wick product can be defined by writing the monomial

X _{1...} X _{k} as a "Wick polynomial":X _{1...} X _{k} = \sum _{S\subseteq\left\{1,...,k\right\} } \operatorname\left(\textstyle\prod_ 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}} , ... ,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

\langle X \rangle = X - \operatorname{E}X, \langle X, Y \rangle = X Y - \operatorname{E}Y ⋅ X - \operatorname{E}X ⋅ Y + 2 (\operatorname{E}X)(\operatorname{E}Y) - \operatorname{E}(X Y), \begin{align} \langle X,Y,Z\rangle = &XYZ\\ & - \operatorname{E}Y ⋅ XZ\\ & - \operatorname{E}Z ⋅ XY\\ & - \operatorname{E}X ⋅ YZ\\ & +2 (\operatorname{E}Y)(\operatorname{E}Z) ⋅ X\\ & +2 (\operatorname{E}X)(\operatorname{E}Z) ⋅ Y\\ & +2 (\operatorname{E}X)(\operatorname{E}Y) ⋅ Z\\ & - \operatorname{E}(XZ) ⋅ Y\\ & - \operatorname{E}(XY) ⋅ Z\\ & - \operatorname{E}(YZ) ⋅ 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{align}

Another notational convention

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

: X _{1,} ... , 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, ... ,X \rangle

with n factors.

The sequence of polynomials P_n such that

P _{n(X)} = \langle X, ... ,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 ^{n} _{i =0} {n\choose i}a ^{ib} ^{n-i} X ^{'i} Y ^{'{n-i} }

Wick exponential

\langle \operatorname{exp}(aX)\rangle \stackrel{ def } \ \sum_^\infty\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.