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,

is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:

(i.e. the empty productthe product of no random variables at allis 1). Thereafter finite moments must be assumed. Next, for k≥1,

where means Xi is absent, and the constraint that

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.

Binomial theorem

Wick exponential

References

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