Wick product

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In probability theory, the Wick product

\langle X_1,\dots,X_k \rangle\,

named after physicist Gian-Carlo Wick, is a sort of product of the random variables, X1, ..., Xk, defined recursively as follows:

\langle \rangle = 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

{\partial\langle X_1,\dots,X_k\rangle \over \partial X_i} = \langle X_1,\dots,X_{i-1}, \widehat{X}_i, X_{i+1},\dots,X_k \rangle,

where \widehat{X}_i means Xi is absent, and the constraint that

E\langle X_1,\dots,X_k\rangle = 0\mbox{ for }k \ge 1.\,

Contents

[edit] Examples

It follows that

\langle X \rangle = X - EX,\,


\langle X, Y \rangle = X Y - EY\cdot X - EX\cdot Y+ 2(EX)(EY) - E(X Y).\,


\begin{align} \langle X,Y,Z\rangle =&XYZ\\ &-EY\cdot XZ\\ &-EZ\cdot XY\\ &-EX\cdot YZ\\ &+2(EY)(EZ)\cdot X\\ &+2(EX)(EZ)\cdot Y\\ &+2(EX)(EY)\cdot Z\\ &-E(XZ)\cdot Y\\ &-E(XY)\cdot Z\\ &-E(YZ)\cdot X\,\\ \end{align}

[edit] 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 of the random variable X.

[edit] Wick powers

The nth Wick power of a random variable X is the Wick product

X'^n = \langle X,\dots,X \rangle\,

with n factors.

The sequence of polynomials Pn such that

P_n(X) = \langle X,\dots,X \rangle = X'^n\,

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

P_n'(x) = nP_{n-1}(x),\,

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

X'^n = B_n(X)\,

where Bn is the nth-degree Bernoulli polynomial.

[edit] Binomial theorem

(aX+bY)^{'n} = \sum_{i=0}^n {n\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}}

[edit] Wick exponential

\langle \operatorname{exp}(aX)\rangle \ \stackrel{\mathrm{def}}{=} \ \sum_{i=0}^\infty\frac{a^i}{i!} X^{'i}

[edit] References

  • [1] 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.