Gaussian/Chi-square expectation for a rational function with high-degree term

Question

Suppose $X \sim N(0,1)$ is a standard Gaussian random variable. How to calculate the following expectation for some $a>0$ and integer $k\ge 1$? $$ \mathbb{E}_X\left[\frac{1}{a + X^{2k}}\right] $$

Answer 1

After some routine scaling substitutions you reduce the problem to finding

$ I(t)= \int \frac{e^{-t u^2}}{ 1+u^{2k}} \ du $.

Note that derivatives with respect to $t$ bring down powers of $u^2$. Thus after $k$ such derivatives you get a numerator term that nearly matches the denominator. In fact you can check that $I(t)$ satisfies the differential equation

$(\frac{ -d}{dt})^k I(t) + I(t) = \int e^{-t u^2} \ du = f(t) =\sqrt{\pi} t^{-1/2}$.

This is a linear constant-coefficient inhomogeneous differential equation with inhomogeneous term $f(t)$. There are many equivalent methods that can be applied to obtain the final solution. They boil down to first solving the homogeneous equation which are of the form $ e^{\omega t}$ in which $\omega$ are chosen to solve the characteristic equation $(-\omega)^k +1=0$ . These are complex roots of $\pm 1$. Then the inhomogeneous equation can be written as a convolution integral formula involving the product of these exponentials and the forcing term. I believe you should finally obtain an answer that is a linear combination of such integrals, each of which has the form Erf$(\omega \sqrt{t})$.

P.S. The coefficients in the linear combination will depend on the initial data which are the values of the derivatives of $I$ at $t=0$. These are simply various Gaussian moments that are well-tabulated.

Note that the exponentials $ e^{\omega t}$ associated to the complex roots $\omega$ will oscillate. Thus they will probably mostly cancel when $t$ is large. In general you should expect the root(s) with the largest real part to dominate.

P.P.S. Also, by expanding $f(t)$ in a generalized Taylor series ( a Puiseux series) of the form $\sum_{n=0}^{\infty} c_{n-1/2}\frac{ t^{n-(1/2)} }{ (n-(1/2))!}$ you can get a nice recurrence relation for the expansion constants. Here fractional factorials are typeset shortcuts for values of the Gamma function.