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, ..., Xk 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&mdash;the product of no random variables at all&mdash;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 Xi 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


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


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



\begin{align} \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{align}$$

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


 * $$ X'^n = H_n(X)\, $$

where Hn is the nth Hermite polynomial.

Binomial theorem

 * $$ (aX+bY)^{'n} = \sum_{i=0}^n {n\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}}$$

Wick exponential

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