Given a random process $X(t)$, it's mean can be found as
$$\langle X(t) \rangle = \lim_{T \rightarrow \infty} \frac{1}{T}\int_0^T X(t) \mathrm{d}t$$
The normal way of estimating this would be to simply take the a finite value of $T$ and then numerically integrate (e.g. left rectangle rule or trapezoid rule). For increasing $T$, the convergence of the estimate for left rectangle rule is $\mathcal{O}(1/\sqrt{T})$ (or something to that effect).
I've been toying around with the idea of:
- Using higher order quadrature methods
- Whether use of higher order quadrature methods increases the accuracy of the estimate and/or improves the convergence of the estimate with respect to $T$
I've tried searching, but haven't found any literature (or anything on the internet) proposing solutions for 1 or an answer for 2.
Specifically, I have two ideas:
Gauss-Laguerre Quadrature
This is a Gauss quadrature that evaluates integrals with a domain of $[0,\infty)$. While this could be used to evaluate
$$ \int_0^\infty X(t) \mathrm{d}t$$
I can't then divide by $1/T$. This in addition to the fact that the integral obviously does not converge without the fraction scaling at the front of it. If there were some way to get the $1/T$ term into the integrand, then maybe this could work?
Composite Gauss Quadrature
This would simply break up some finite span $[0,T)$ into a series of pieces, then use "standard" Gauss-Legendre quadrature. Obviously the integral will be more accurate, but whether it improves the mean estimate, I'm not sure.
Disclaimer: I know that the order accuracy result of Gaussian quadrature (and others) comes from refinement of a fixed domain rather than improved convergence for increasing $T$. I'm just wondering if there's been any results for problems like this
