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| In [[probability theory]], the '''Wick product''' is a particular way of defining an adjusted [[product (mathematics)|product]] of a set of [[random variable]]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.
| | Hi there, I am Andrew Berryhill. Invoicing is my occupation. One of the very very best things in the globe for him is doing ballet and he'll be starting something else alongside with it. For a whilst I've been in Alaska but I will have to move in a year or two.<br><br>Feel free to visit my web blog; spirit messages ([http://www.octionx.sinfauganda.co.ug/node/22469 please click for source]) |
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| 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.
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| The Wick product is named after physicist [[Gian-Carlo Wick]].
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| ==Definition==
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| The Wick product,
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| :<math>\langle X_1,\dots,X_k \rangle\,</math>
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| is a sort of [[product (mathematics)|product]] of the [[random variable]]s, ''X''<sub>1</sub>, ..., ''X''<sub>''k''</sub>, defined recursively as follows:{{Citation needed|date=May 2012}}
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| :<math>\langle \rangle = 1\,</math>
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| (i.e. the [[empty product]]—the product of no random variables at all—is 1). Thereafter finite [[moment (mathematics)|moments]] must be assumed. Next, for ''k''≥1,
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| :<math>{\partial\langle X_1,\dots,X_k\rangle \over \partial X_i}
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| = \langle X_1,\dots,X_{i-1}, \widehat{X}_i, X_{i+1},\dots,X_k \rangle,</math>
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| where <math>\widehat{X}_i</math> means ''X''<sub>''i''</sub> is absent, and the constraint that
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| :<math> \operatorname{E} \langle X_1,\dots,X_k\rangle = 0\mbox{ for }k \ge 1. \,</math>
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| ==Examples==
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| It follows that
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| :<math>\langle X \rangle = X - \operatorname{E}X,\,</math>
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| :<math>\langle X, Y \rangle = X Y - \operatorname{E}Y\cdot X - \operatorname{E}X\cdot Y+ 2(\operatorname{E}X)(\operatorname{E}Y) - \operatorname{E}(X Y).\,</math>
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| :<math>
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| \begin{align}
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| \langle X,Y,Z\rangle
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| =&XYZ\\
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| &-\operatorname{E}Y\cdot XZ\\
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| &-\operatorname{E}Z\cdot XY\\
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| &-\operatorname{E}X\cdot YZ\\
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| &+2(\operatorname{E}Y)(\operatorname{E}Z)\cdot X\\
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| &+2(\operatorname{E}X)(\operatorname{E}Z)\cdot Y\\
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| &+2(\operatorname{E}X)(\operatorname{E}Y)\cdot Z\\
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| &-\operatorname{E}(XZ)\cdot Y\\
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| &-\operatorname{E}(XY)\cdot Z\\
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| &-\operatorname{E}(YZ)\cdot X\,\\
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| \end{align}</math>
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| <!--
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| Perhaps the next several could be added here. -->
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| ==Another notational convention==
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| In the notation conventional among physicists, the Wick product is often denoted thus:
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| :<math>: X_1, \dots, X_k:\,</math>
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| and the angle-bracket notation
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| :<math>\langle X \rangle\,</math>
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| is used to denote the [[expected value]] of the random variable ''X''.
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| ==Wick powers==
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| The ''n''th '''Wick power''' of a random variable ''X'' is the Wick product
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| :<math>X'^n = \langle X,\dots,X \rangle\,</math>
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| with ''n'' factors.
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| The sequence of polynomials ''P''<sub>''n''</sub> such that
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| :<math>P_n(X) = \langle X,\dots,X \rangle = X'^n\,</math>
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| form an [[Appell sequence]], i.e. they satisfy the identity
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| :<math>P_n'(x) = nP_{n-1}(x),\,</math>
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| for ''n'' = 0, 1, 2, ... and ''P''<sub>0</sub>(''x'') is a nonzero constant.
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| For example, it can be shown that if ''X'' is [[uniform distribution (continuous)|uniformly distributed]] on the interval [0, 1], then
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| :<math> X'^n = B_n(X)\, </math>
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| where ''B''<sub>''n''</sub> is the ''n''th-degree [[Bernoulli polynomial]]. Similarly, if ''X'' is [[Normal distribution|normally distributed]] with variance 1, then
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| :<math> X'^n = H_n(X)\, </math>
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| where ''H''<sub>''n''</sub> is the ''n''th [[Hermite polynomial]].
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| ==Binomial theorem==
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| :<math> (aX+bY)^{'n} = \sum_{i=0}^n {n\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}}</math>
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| == Wick exponential ==
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| :<math>\langle \operatorname{exp}(aX)\rangle \ \stackrel{\mathrm{def}}{=} \ \sum_{i=0}^\infty\frac{a^i}{i!} X^{'i}</math>
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| {{No footnotes|date=May 2012}}
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| ==References==
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| * [http://www.encyclopediaofmath.org/index.php/Wick_product Wick Product] ''Springer Encyclopedia of Mathematics''
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| * Florin Avram and [[Murad Taqqu]], (1987) "Noncentral Limit Theorems and Appell Polynomials", ''Annals of Probability'', volume 15, number 2, pages 767—775, 1987.
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| * 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
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| * Wick, G. C. (1950) "The evaluation of the collision matrix". ''Physical Rev.'' 80 (2), 268–272.
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| * Hu, Yao-zhong; Yan, Jia-an (2009) [http://arxiv.org/abs/0901.4911v1 "Wick calculus for nonlinear Gaussian functionals"], ''Acta Mathematicae Applicatae Sinica (English Series)'', 25 (3), 399–414 {{doi|10.1007/s10255-008-8808-0}}
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| [[Category:Algebra of random variables]]
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Hi there, I am Andrew Berryhill. Invoicing is my occupation. One of the very very best things in the globe for him is doing ballet and he'll be starting something else alongside with it. For a whilst I've been in Alaska but I will have to move in a year or two.
Feel free to visit my web blog; spirit messages (please click for source)