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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.