Simplify multiple integration of an exponential function

Question

I need to simplify the following integration: $I=\int_a^b\int_a^b...\int_a^b\exp\{-\frac{1}{2}x^TAx\}dx_1dx_2...dx_n$, where $x=(x_1,x_2,...,x_n)^T$ and $A$ is a positive definite matrix of size $n$, $a,b$ - fixed real numbers. Is there any way to make it simple? Thanks !

Answer 1

We can try to use the rule for multidimensional substitution to make this integral look simpler (Spoiler alert: This will not result in a solvable integral, however I'm going through the steps for the substitution anyway):

$$\int_{\varphi(U)} f(\mathbf{x})\, d \mathbf{x} = \int_U f(\varphi(\mathbf{y})) \left|\det(\operatorname{D}\varphi)(\mathbf{y})\right| \,d \mathbf{y}$$

Here, we're looking for a function $\varphi$ which simplifies $f(\mathbf{x}) = \exp(-\mathbf{x}^TA\mathbf{x}/2)$ to $f(\varphi(\mathbf y))= \exp(-\mathbf{y}^T \mathbf{y}/2)$.

Staring at this formula suggests $\mathbf y = A^{1/2} \mathbf x$, and in fact $A^{1/2}$ is well-defined since $A$ is symmetric (and hence diagonaliable) and all of its eigenvalues are positive. Positivity ensures that this matrix is invertible and we get $\varphi(\mathbf y) = A^{-1/2} \mathbf y$ and hence $$f(\varphi(\mathbf y))=\exp\left(-\frac{\mathbf{y^T} A^{-1/2}AA^{-1/2}\mathbf y }{2}\right)= \exp\left(-\frac{\mathbf{y^T}\mathbf y }{2}\right).$$

Since $\varphi$ is linear, its derivative is constant and precisely its representing matrix, i.e. $$\left|\det(\operatorname{D}\varphi)(\mathbf{y})\right|=|\det A^{-1/2}|=\left( \prod_{i=1}^n \lambda_i \right)^{-1/2}$$ where $\lambda_i$ are the eigenvalues of $A$ (and the determinant is the product of the inverse square roots of them).

Finally for the domain of integration, we are given $\varphi(U) = [a,b]^n$, i.e.

$U = A^{1/2}([a,b]^n)$.

To put it together, our integral now looks like

$$\left(\prod_{i=1}^n \lambda_i \right)^{-1/2} \int_{A^{1/2}([a,b]^n)}\exp\left(-\frac{\mathbf y ^T \mathbf y}{2}\right) d \mathbf y.$$

This is slightly nicer than the original integral, but does remind of the distribution function of a standard normal distribution $\exp(-x^2/2)$. In fact, we are trying to compute a probability for a multivariate normal distribution. Unfortunately, no nice substitution can help us to solve this integral analytically.