Wick product

In probability theory, the Wick product

:$langle\; X\_1,dots,X\_k\; angle,$

named after physicist Gian-Carlo Wick, is a sort of product of the random variables, "X"₁, ..., "X"_"k", defined recursively as follows:

:$langle\; angle\; =\; 1,$

(i.e. the empty product—the product of no random variables at all—is 1). Thereafter we must assume finite moments. Next we have

:$\{partiallangle\; X\_1,dots,X\_k\; angle\; over\; partial\; X\_i\}=\; langle\; X\_1,dots,X\_\{i-1\},\; widehat\{X\}\_i,\; X\_\{i+1\},dots,X\_k\; angle,$

where $widehat\{X\}\_i$ means "X"_"i" is absent, and the constraint that

:$Elangle\; X\_1,dots,X\_k\; angle\; =\; 0mbox\{\; for\; \}k\; ge\; 1.,$

Examples

It follows that

:$langle\; X\; angle\; =\; X\; -\; EX,,$

:$langle\; X,\; Y\; angle\; =\; X\; Y\; -\; EYcdot\; X\; -\; EXcdot\; Y+\; 2(EX)(EY)\; -\; E(X\; Y).,$

:

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

:$langle\; X\; angle,$

is used to denote the expected value of the random variable "X".

Wick powers

The "n"th Wick power of a random variable "X" is the Wick product

:$X\text{'}^n\; =\; langle\; X,dots,X\; angle,$

with "n" factors.

The sequence of polynomials "P"_"n" such that

:$P\_n(X)\; =\; langle\; X,dots,X\; angle\; =\; X\text{'}^n,$

form an Appell sequence, i.e. they satisfy the identity

:$P\_n\text{'}(x)\; =\; nP\_\{n-1\}(x),,$

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 [0, 1] , then

:$X\text{'}^n\; =\; B\_n(X),$

where "B"_"n" is the "n"th-degree Bernoulli polynomial.

Binomial theorem

:$(aX+bY)^\{\text{'}n\}\; =\; sum\_\{i=0\}^n\; \{nchoose\; i\}a^ib^\{n-i\}\; X^\{\text{'}i\}\; Y^\{\text{'}\{n-i$

Wick exponential

:$langle\; operatorname\{exp\}(aX)\; angle\; stackrel\{mathrm\{def\{=\}\; sum\_\{i=0\}^inftyfrac\{a^i\}\{i!\}\; X^\{\text{'}i\}$

References

* [http://eom.springer.de/w/w097870.htm] "Springer Encyclopedia of Mathematics"
* Florin Avram and Murad Taqqu, "Noncentral Limit Theorems and Appell Polynomials", "Annals of Probability", volume 15, number 2, pages 767—775, 1987.