Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, with an unknown mean $\mu = (x_0,y_0)$ where $x_0$ and $y_0$ are real number coordinates.
Consider the problem of estimating $\mu$ by drawing $N$ iid random real number variates from the probability distribution (corresponding to real number two-dimensional coordinates), then rounding these variates to the nearest integer coordinate. For example, we may draw a real valued coordinate $(10.49344..., 2.553244)$ according to the described PDF, and we would then report this coordinate as $(10, 3)$.
Let $N$ be the number of sampling events. As a function of the $\sigma_x \approx \sigma_y$, i.e. the standard deviation parameters for our bivariate probability distribution for the particle's position, and the real number valued mean of the particle's position, $\mu$, how well can we estimate $\mu$?
Of course, if $\sigma_x$ and $\sigma_y$ are $<<1$, and depending on the difference between $\mu$ and the nearest integer coordinate, the error can be quite significant for small $N$. I'm curious how the required value of $N$ scales for localizing $\mu$ with some error $\epsilon$.
