Wick product

In probability theory, the Wick product


 * $$\langle X_1,\dots,X_k \rangle\,$$

named after physicist Gian-Carlo Wick, is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:


 * $$\langle \rangle = 1\,$$

(i.e. the empty product&mdash;the product of no random variables at all&mdash;is 1). Thereafter we must assume finite moments. Next we have


 * $${\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


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

Examples
It follows that


 * $$\langle X \rangle = X - EX,\,$$


 * $$\langle X, Y \rangle = X Y - EY\cdot X - EX\cdot Y+ 2(EX)(EY) - E(X Y).\,$$



\begin{align} \langle X,Y,Z\rangle =&XYZ\\ &-EY\cdot XZ\\ &-EZ\cdot XY\\ &-EX\cdot YZ\\ &+2(EY)(EZ)\cdot X\\ &+2(EX)(EZ)\cdot Y\\ &+2(EX)(EY)\cdot Z\\ &-E(XZ)\cdot Y\\ &-E(XY)\cdot Z\\ &-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.

Binomial theorem

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

Wick exponential

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