Isserlis' theorem

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In probability theory, Isserlis’ theorem or Wick’s theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.

This theorem is particularly important in particle physics, where it is known as Wick's theorem after the work of Wick (1950). Other applications include the analysis of portfolio returns,[1] quantum field theory[2] and generation of colored noise.[3]

Theorem statement

If (x1, …, x2n) is a zero mean multivariate normal random vector, then

\begin{align} & \operatorname{E} [\,x_1x_2\cdots x_{2n}\,] = \sum\prod \operatorname{E}[\,x_ix_j\,], \\ & \operatorname{E}[\,x_1x_2\cdots x_{2n-1}\,] = 0, \end{align}

where the notation ∑ ∏ means summing over all distinct ways of partitioning x1, …, x2n into pairs.[4] This yields $(2n)!/(2^{n}n!)$ terms in the sum. For example, for fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms (as you can check in the examples below).

In his original paper,[5] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the fourth-order moments,[6] which takes the appearance

$\operatorname{E}[\,x_1x_2x_3x_4\,] = \operatorname{E}[x_1x_2]\,\operatorname{E}[x_3x_4] + \operatorname{E}[x_1x_3]\,\operatorname{E}[x_2x_4] + \operatorname{E}[x_1x_4]\,\operatorname{E}[x_2x_3].$

Another example of Isserlis' theorem is:

\begin{align} & {} \operatorname{E}[X_1 X_2 X_3 X_4 X_5 X_6] \\ &{} = \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_4 ]\operatorname{E}[X_5 X_6 ] + \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_5 ]\operatorname{E}[X_4 X_6] + \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_6 ]\operatorname{E}[X_4 X_5] \\ &{} + \operatorname{E}[X_1 X_3 ]\operatorname{E}[X_2 X_4 ]\operatorname{E}[X_5 X_6 ] + \operatorname{E}[X_1 X_3 ]\operatorname{E}[X_2 X_5 ]\operatorname{E}[X_4 X_6 ] + \operatorname{E}[X_1 X_3]\operatorname{E}[X_2 X_6]\operatorname{E}[X_4 X_5] \\ &+ \operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_3]\operatorname{E}[X_5 X_6]+\operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_5]\operatorname{E}[X_3 X_6]+\operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_6]\operatorname{E}[X_3 X_5] \\ & + \operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_3]\operatorname{E}[X_4 X_6]+\operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_4]\operatorname{E}[X_3 X_6]+\operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_6]\operatorname{E}[X_3 X_4] \\ & + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_3]\operatorname{E}[X_4 X_5 ] + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_4 ]\operatorname{E}[X_3 X_5] + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_5]\operatorname{E}[X_3 X_4]. \end{align}