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 product is named after physicist Gian-Carlo Wick, cf. Wick's theorem.

Definition[edit]

The Wick product,

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

(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 means Xi is absent, and the constraint that

Examples[edit]

It follows that

Another notational convention[edit]

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[edit]

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[edit]

Wick exponential[edit]

References[edit]

  • 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.
  • Hu, Yao-zhong; Yan, Jia-an (2009) "Wick calculus for nonlinear Gaussian functionals", Acta Mathematicae Applicatae Sinica (English Series), 25 (3), 399–414 doi:10.1007/s10255-008-8808-0