Structure tensor

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.

Definition

The Wick product,

$⟨X_1,\dots ,X_k⟩$

is a sort of product of the random variables, X₁, ..., X_k, defined recursively as follows:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

$⟨⟩=1$

(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,

$\frac{\partial ⟨X_1,\dots ,X_k⟩}{\partial X_i}=⟨X_1,\dots ,X_{i-1},{\hat{X}}_i,X_{i+1},\dots ,X_k⟩,$

where ${\hat{X}}_i$ means X_i is absent, and the constraint that

$\mathrm{E}⟨X_1,\dots ,X_k⟩=0\text{ for }k\ge 1.$

Examples

It follows that

$⟨X⟩=X-\mathrm{E}X,$

$⟨X,Y⟩=XY-\mathrm{E}Y\cdot X-\mathrm{E}X\cdot Y+2(\mathrm{E}X)(\mathrm{E}Y)-\mathrm{E}(XY).$

$\begin{array}{rl}⟨X,Y,Z⟩=& XYZ\\ & -\mathrm{E}Y\cdot XZ\\ & -\mathrm{E}Z\cdot XY\\ & -\mathrm{E}X\cdot YZ\\ & +2(\mathrm{E}Y)(\mathrm{E}Z)\cdot X\\ & +2(\mathrm{E}X)(\mathrm{E}Z)\cdot Y\\ & +2(\mathrm{E}X)(\mathrm{E}Y)\cdot Z\\ & -\mathrm{E}(XZ)\cdot Y\\ & -\mathrm{E}(XY)\cdot Z\\ & -\mathrm{E}(YZ)\cdot X\\ \end{array}$

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

$⟨X⟩$

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

Wick powers

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

$X{\text{'}}^n=⟨X,\dots ,X⟩$

with n factors.

The sequence of polynomials P_n such that

$P_n(X)=⟨X,\dots ,X⟩=X{\text{'}}^n$

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

${P_n}^{\prime }(x)=n{P}_{n-1}(x),$

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

$X{\text{'}}^n=B_n(X)$

where B_n is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then

$X{\text{'}}^n=H_n(X)$

where H_n is the nth Hermite polynomial.

Binomial theorem

$(aX+bY{)}^{\text{'}n}=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})a^i{b}^{n-i}{X}^{\text{'}i}{Y}^{\text{'}n-i}$

Wick exponential

$⟨\exp (aX)⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _{i=0}^{\mathrm{\infty }}\frac{a^i}{i!}{X}^{\text{'}i}$

References

• Wick Product Springer Encyclopedia of Mathematics

• Florin Avram and Murad Taqqu, (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.

• Hu, Yao-zhong; Yan, Jia-an (2009) "Wick calculus for nonlinear Gaussian functionals", Acta Mathematicae Applicatae Sinica (English Series), 25 (3), 399–414 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.