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, ..., Xk, 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 Xi is absent, and the constraint that

\operatorname{E} \langle X_1,\dots,X_k\rangle = 0\mbox{ for }k \ge 1. \,

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\,\\ \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}

References