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.
Contents
- Definition
- Examples
- Another notational convention
- Wick powers
- Binomial theorem
- Wick exponential
- References
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,
is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:
(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,
where
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 variable X.
Wick powers
The nth Wick 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(x) is a nonzero constant.
For example, it can be shown that if X is uniformly distributed on the interval [0, 1], 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.