TheInfoList

In
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
, the Wick product is a particular way of defining an adjusted product of a set of
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s. 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 polynomial In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in th ...
s or
Hermite polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s. The Wick product is named after physicist Gian-Carlo Wick, cf.
Wick's theorem Wick's theorem is a method of reducing high- order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilat ...
.

Definition

Assume that ''X''1, ..., ''X''''k'' are
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s with finite moments. The Wick product :$\langle X_1,\dots,X_k \rangle\,$ is a sort of product defined recursively as follows: :$\langle \rangle = 1\,$ (i.e. the
empty product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
—the product of no random variables at all—is 1). For ''k'' ≥ 1, we impose the requirement :$= \langle X_1,\dots,X_, \widehat_i, X_,\dots,X_k \rangle,$ where $\widehat_i$ means that ''X''''i'' is absent, together with the constraint that the average is zero, :$\operatorname \langle X_1,\dots,X_k\rangle = 0. \,$

Examples

It follows that :$\langle X \rangle = X - \operatornameX,\,$ :$\langle X, Y \rangle = X Y - \operatornameY\cdot X - \operatornameX\cdot Y+ 2\left(\operatornameX\right)\left(\operatornameY\right) - \operatorname\left(X Y\right).\,$ :$\begin \langle X,Y,Z\rangle =&XYZ\\ &-\operatornameY\cdot XZ\\ &-\operatornameZ\cdot XY\\ &-\operatornameX\cdot YZ\\ &+2\left(\operatornameY\right)\left(\operatornameZ\right)\cdot X\\ &+2\left(\operatornameX\right)\left(\operatornameZ\right)\cdot Y\\ &+2\left(\operatornameX\right)\left(\operatornameY\right)\cdot Z\\ &-\operatorname\left(XZ\right)\cdot Y\\ &-\operatorname\left(XY\right)\cdot Z\\ &-\operatorname\left(YZ\right)\cdot X\\ &-\operatorname\left(XYZ\right)\\ &+2\operatorname\left(XY\right)\operatornameZ+2\operatorname\left(XZ\right)\operatornameY+2\operatorname\left(YZ\right)\operatornameX\\ &-3\left(\operatornameX\right)\left(\operatornameY\right)\left(\operatornameZ\right). \end$

Another notational convention

In the notation conventional among physicists, the Wick product is often denoted thus: :$: X_1, \dots, X_k:\,$ and the angle-bracket notation :$\langle X \rangle\,$ is used to denote the
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
of the random variable ''X''.

Wick powers

The ''n''th Wick power of a random variable ''X'' is the Wick product :$X\text{'}^n = \langle X,\dots,X \rangle\,$ with ''n'' factors. The sequence of polynomials ''P''''n'' such that :$P_n\left(X\right) = \langle X,\dots,X \rangle = X\text{'}^n\,$ form an
Appell sequence In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity :\frac p_n(x) = np_(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the tri ...
, i.e. they satisfy the identity :$P_n\text{'}\left(x\right) = nP_\left(x\right),\,$ for ''n'' = 0, 1, 2, ... and ''P''0(''x'') is a nonzero constant. For example, it can be shown that if ''X'' is uniformly distributed on the interval , 1 then :$X\text{'}^n = B_n\left(X\right)\,$ where ''B''''n'' is the ''n''th-degree
Bernoulli polynomial In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in th ...
. Similarly, if ''X'' is
normally distributed In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

with variance 1, then :$X\text{'}^n = H_n\left(X\right)\,$ where ''H''''n'' is the ''n''th
Hermite polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

.

Binomial theorem

:$\left(aX+bY\right)^ = \sum_^n a^ib^ X^ Y^$

Wick exponential

:$\langle \operatorname\left(aX\right)\rangle \ \stackrel \ \sum_^\infty\frac X^$ {{No footnotes, date=May 2012

References

Wick Product
''Springer Encyclopedia of Mathematics'' * Florin Avram and
Murad Taqqu Murad Taqqu (Arabic language, Arabic: مراد تقّو) is an Iraqi probabilist and statistician specializing in time series and stochastic processes. His research areas have included Long-range dependency, long-range dependence, self-similar proce ...
, (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. Algebra of random variables