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Wick product

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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 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

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Assume that X1, ..., Xk are random variables with finite moments. The Wick product

is a sort of product defined recursively as follows:[citation needed]

(i.e. the empty product—the product of no random variables at all—is 1). For k ≥ 1, we impose the requirement

where means that Xi is absent, together with the constraint that the average is zero,

Equivalently, the Wick product can be defined by writing the monomial as a "Wick polynomial":

,

where denotes the Wick product if . This is easily seen to satisfy the inductive definition.

Examples

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It follows that

Another notational convention

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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

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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

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Wick exponential

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References

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  • Wick Product Springer Encyclopedia of Mathematics
  • Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages 767—775, 1987.
  • Hida, T. and Ikeda, N. (1967) "Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral". Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press
  • Wick, G. C. (1950) "The evaluation of the collision matrix". Physical Rev. 80 (2), 268–272.