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 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 is a sort of product defined recursively as follows: . For k ≥ 1, we impose the requirement where means that Xi is absent, together with the constraint that the average is zero,
Examples
It follows that
Another notational convention
In the notation conventional among physicists, the Wick product is often denoted thus: and the angle-bracket notation is used to denote the expected value of the random variableX.
Wick powers
The nthWick power of a random variable X is the Wick product with n factors. The sequence of polynomials Pn such that form an Appell sequence, i.e. they satisfy the identity for n = 0, 1, 2,... and P0 is a nonzero constant. For example, it can be shown that if X is uniformly distributed on the interval , then where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then where Hn is the nth Hermite polynomial.
