Note: I originally tried to pose this question generally, without discussing the specific type of stochastic process. I hope that this can still be an interesting question generally.
Assume that we have a point process $\Phi$ with support on some bounded complete space $B\subset\mathbb{R}$. Assume that $\Phi$ is characterized by the intensity measure $\Lambda(\cdot)$, which has intensity function $\lambda(\cdot)$. This means that $\Lambda(B)=\int_B \lambda(x)dx=\mathbb{E}[\Phi(B)]$.
Now that the necessary notation has been introduced, here is my question: suppose that I observe a realization of $\Phi$, which we write as a set $\phi=\{x_1,\ldots,x_n\}$. Further suppose that we believe that the intensity function (which we can think of as the data generating process for $\phi$) is of the form $f(x,\alpha)=\sum_{i=0}^k\alpha_i x^i$.
If we then plot a histogram of $\phi$, can we "recover'' the coefficients $\alpha_i$ from this histogram, up to some closeness? More formally if $n$ gets very large, and the number of bins $b$ gets very large (or equivalently, the bin width goes to 0), can we fit some polynomial $ f(x,\hat\alpha)=\sum_{i=0}^k\hat\alpha_i x^i$ from the histogram such that $\forall \epsilon>0$, $|\alpha_i-\hat\alpha_i|<\epsilon$?
This is basically a density fitting problem. Intuitively, I think that given enough data, we should be able to recover the coefficients. I am not interested in a kernel density estimation (KDE) approach, rather just if there is some theorem, or paper that you are aware of that demonstrates this result.
Please note I also asked the exact same question on Cross Validated Stack Exchange .