I learnt about Isserlis’ theorem (also known as Wick’s probability theorem) at a talk today. The theorem comes from a paper from 1918, which is listed as Reference 1 below. In the words of Reference 2, the theorem
… allows [us] express to the expectation of a monomial in an arbitrary number of components of a zero mean Gaussian vector in terms of the entries of its covariance matrix only.
We introduce some notation (as in Reference 2) to state the theorem succinctly. Let be a set of integers such that for all . The need not be distinct. For any vector , denote
with the convention that . Let denote the set of all pairings of , i.e. partitions of into disjoint pairs. For a pairing , let denote the set of indices such that the pairs in are .
(As an example, if , one possible pairing is . For this pairing, a possible choice of is , with and .)
We are now ready to state the theorem:
Theorem (Isserlis’ theorem): Let be a set of integers such that for all , and let be a Gaussian vector with zero mean. If is even, then
If is odd, then .
Here are some special cases of Isserlis’ theorem to demonstrate how to interpret the equation above. If for , there are 3 possible pairings, giving us
If we take for , there are still 3 possible pairings, and we get
This tells us that the 4th moment of a mean-zero 1-dimensional gaussian random variable is 3 times the square of its 2nd moment.
As a final example, if we take and , we still have 3 possible pairings, and we get