# Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$.

Consider also its KL expansion $X(t) = \sum\limits_{k=0}^{\infty} Z_k e_k (t)$, with $Z_k$ being pairwise uncorrelated Gaussian random variables.

I'm interested in what is known about the convergence rate of the finite expansion $\sum\limits_{k=0}^N Z_k e_k (t)$. Mainly

• When can we expect an exponentialy small pointwise (or a.e.) error term in $N$?
• Given a smooth function $f$, when can we approximate $\int\limits_{\mathbb{R}} f(X(t)) \, dt$ with an integration $d^N \mathbf{Z}$ with exponentially small error term.

Extra Details: I've learned that the functions $e_k (t)$ are, in fact, the eigenvectors of the following integral Kernel over $L^2\left(\mathbb{R} \right)$, $$Ku(t) = \int\limits_{\mathbb{R}} \mathbb{E} \left[ X_t X_s \right] u(s) \, ds \, .$$

For more details, see this coincise introductory here.

Thanks