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Wick product : ウィキペディア英語版
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,
:\langle X_1,\dots,X_k \rangle\,
is a sort of product of the random variables, ''X''1, ..., ''X''''k'', defined recursively as follows:
:\langle \rangle = 1\,
(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,
:
= \langle X_1,\dots,X_, \widehat_i, X_,\dots,X_k \rangle,
where \widehat_i means ''X''''i'' is absent, and the constraint that
: \operatorname \langle X_1,\dots,X_k\rangle = 0\mboxk \ge 1. \,

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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