Multivariate gaussian integral

[Alexandra97]

I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $\times$ 1 and T is a positive definite symmetric p $\times$ p matrix. The integral is the following:

$$\int_{x=-\infty}^{x=a} x e^{ (-0.5 x'T^{-1}x)} d^p\bf{x}$$

I have tried to solve it with the cholesky decomposition and substitution with the Jacobian, but the dimension of the solution is not a p $\times$ 1 vector. When a equals $\infty$ the solution is of course an p $\times$ 1 vector of zero's (since it is the expected value).

Since I am not very experienced in integrating over vectors, also a good reference about this subject would be greatly appreciated.

 

[jvicens]

Thank you for your question regarding the multivariate gaussian integral. This type of integral is commonly encountered in many fields of science and mathematics, and there are several methods for solving it.

One approach to solving this integral is by using the Cholesky decomposition, as you have mentioned. This method involves decomposing the covariance matrix T into a lower triangular matrix L such that T = LL'. Then, by making a change of variables x = Ly, the integral can be transformed into a standard multivariate gaussian integral with a diagonal covariance matrix. This can be solved using standard methods, and the resulting solution will be a p $\times$ 1 vector.

Another approach is by using the spectral decomposition of T. This involves diagonalizing T into a matrix UDU', where U is an orthogonal matrix and D is a diagonal matrix containing the eigenvalues of T. The integral can then be transformed into a standard multivariate gaussian integral with a diagonal covariance matrix by making a change of variables x = Uy. Again, this can be solved using standard methods, and the resulting solution will be a p $\times$ 1 vector.

It is also worth mentioning that there are numerical methods for solving this type of integral, such as Monte Carlo integration or numerical quadrature methods. These methods may be useful if an analytical solution cannot be found.

As for references, a good textbook on multivariate calculus or probability theory may provide more detailed explanations and examples of how to solve this type of integral. Some recommended textbooks include "Multivariate Calculus" by James Stewart and "Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker.

I hope this helps and good luck with your integration!