Seminar Talk: Wick Formula for Quaternion Normal Laws

This is a seminar talk of February 13, 2008. 1. Wick’s theorem for R-valued normal vectors The following is known as Wick’s theorem. Theorem A (Wick [9]). If (X1, . . . , X2n) is multivariate normal with mean zero, then E(X1X2 . . . X2n) = ∑ V ∏ {j,k}∈V E(XjXk), where the sum is taken over all pair partitions V of {1, 2, . . . , 2n}.1 For example, E(X1X2X3X4) = E(X1X2)E(X3X4)+E(X1X3)E(X2X4)+E(X1X4)E(X2X3) as {1, 2, 3, 4} has three pair partitions V1 = {{1, 2}, {3, 4}}, V1 = {{1, 3}, {2, 4}}, V1 = {{1, 4}, {3, 2}}. In particular, if X1 = X2 = X3 = X4 = X and E(X) = 1, then the formula gives E(X) = 3. Proof. (This is a consequence of moments-cumulants relation [4]; the connection is best visible in the partition formulation of [7]. For another proof, see [2, page 12].) Suppose first that X1 . . . Xn are of very special form. Namely, suppose they are selected with repetition from a fixed finite i.i.d. N(0, 1) family Z1, Z2, . . . . Then E(X1 . . . Xn) is zero if some of the Z ′s enter the product an odd number of times. If all Z’s appear an even number of times, then the answer is E(X1 . . . Xn) = (2n1−1)!!(2n2−1)!! . . . (2nk−1), where say Z1 is repeated 2n1 times, Z2 is repeated 2n2 times, etc. This matches the answer from the pair partitions: the only contributing partitions are those that pair {1, . . . , 2n1} within itself, and {n1 + 1, . . . , n1 + n2} within itself, etc. The number of pairings of {1, . . . , 2n} is (2n − 1) matches for 1 times the number of pairings of the remaining 2n − 2 elements. Thus it is (2n− 1)× (2n− 3)× . . . 3× 1. Suppose now that we have general multivariate normal Xj = ∑ j Ai,jZj . Then Ca,b = E(XaXb) = ∑ Aa,jAb,j . Date: Created: November 16, 2007. Printed: February 12, 2008 File: quaaternion-wick-08.tex. 1That is, partitions into two-element sets, so each V has the form V = {{j1, k1}, {j2, k2}, . . . , {jn, kn}} .