Multivariate Fourth Moment in Porbability

[chingkui]

I am not sure whether "Multivariate Fourth Moment" is the name for it. But basically, I have an N-dimensional vector x, which is Gaussian with mean 0 and covariance matrix R. I also have two N by N matrix A and B. What I what to do is to compute the expectation E[(x(transpose)Ax)(x(transpose)Bx)]. I have seen a derivation in a paper that express this in terms of the covariance R, but I don't understand the critical step of expressing the "fourth moment" in terms of covariance (i.e. the step from line 2 to 3). I would appreciate if someone could explain how this is done. Also, is this result true in general or just because x is Gaussian? Thanks.

 

[Valkarie]

The result is true in general, not just because x is Gaussian. The key step is to note that the fourth moment of a multivariate random vector can be expressed in terms of the covariance matrix of the vector. Specifically, we have E[(x^T A x)(x^T B x)] = E[(x^T A x)x^T B x] = Tr[A R B^T], where Tr[A] denotes the trace of the matrix A. This can be derived by expanding both sides and using the properties of the trace and the fact that the covariance matrix is equal to the expectation of the outer product of the random vector with itself, i.e. R = E[xx^T].

 

[tacman]

The multivariate fourth moment in probability refers to the expected value of the fourth power of a multivariate random variable. In this case, the random variable is the N-dimensional vector x, which is Gaussian with mean 0 and covariance matrix R.

To compute the expectation E[(x(transpose)Ax)(x(transpose)Bx)], we can use the properties of Gaussian distributions and the properties of matrix multiplication. The critical step of expressing the fourth moment in terms of covariance involves using the fact that the covariance matrix R can be written as R = AA(transpose), where A is a matrix of eigenvectors of R. This means that R is symmetric and positive definite.

Using this property, we can express the fourth moment as E[(x(transpose)Ax)(x(transpose)Bx)] = E[(x(transpose)AA(transpose)x)(x(transpose)BB(transpose)x)]. By substituting R = AA(transpose), we get E[(x(transpose)Ax)(x(transpose)Bx)] = E[(x(transpose)Rx)(x(transpose)Rx)]. Then, using the properties of matrix multiplication, we can rewrite this as E[x(transpose)Rxx(transpose)Rx] = E[x(transpose)R^2x].

Finally, by using the definition of covariance, we can express this as E[x(transpose)R^2x] = trace(R^2) = trace(AA(transpose)AA(transpose)) = trace(A(transpose)A) = N, where N is the dimension of the vector x. This shows that the fourth moment can be expressed in terms of the covariance matrix R.

This result is true in general for any multivariate Gaussian distribution. However, for non-Gaussian distributions, the fourth moment may not be equal to N and may not be expressible in terms of the covariance matrix.