I'm looking for bounds over the following Monte Carlo approximation.
Define a subset $\Omega$ of R^n (compact, not sure if it is important) and probability density function $P(X)$ over it. Further, define a positive-definite kernel $k(X,X')$ over $\Omega$. Last, define some integrable function $f(X)$ over $\Omega$.
I want to approximate the integral transform w.r.t. $k$:
$h(X') = \int_{\Omega} k(X',X) \cdot P(X) \cdot f(X) dX$
with the average:
$\hat{h}(X') = \frac{1}{N} \sum_i k(X',X_i) \cdot f(X_i)$
where $\{ X_i \}$ are $N$ i.i.d. samples from $P$.
How close are functions $h$ and $\hat{h}$? By the closeness I refer to any possible metric over the function space. $L_p^q$ or RKHS norm for example. Likewise, point-wise distance $|h(X) - \hat{h}(X)|$ and its bounds would also help me in my analysis.
I'm mostly interested to relate the above distance to the properties of $k$. What can we say about it based on properties of $k$ and its eigenvalues/eigenfunctions? I know, based on intuition, that for bell-like kernels (e.g. Gaussian) wider bandwidth of the kernel make the above approximation easier/more accurate. Is it proved/discussed by any scientific work? How about the general kernel $k(X',X) = \phi(X')^T \cdot \phi(X)$? What can we say then?
Also, based on $N$ and properties of $f$, what can we say? Does $f$ has to be continuous/differentiable/smooth/infinitely differentiable (or not)? Can the above question be related to the frequency domain of $f$? Likewise, if $f$ is assumed to be within RKHS defined by $k$, how RKHS norm of $f$ affect the distance between $h$ and $\hat{h}$?
I know that I can apply here law of large numbers and similar theorems. Yet, I'm looking for some works/papers that already addressed the actual above setting - the approximation of the integral transform.
Thanks for any feedback.