# Convergence of sample mean of Cartesian product of 1-d samples

I want to estimate the mean of some quantity over a two-dimensional space, say the average height of some bounded geographic area.

it's convenient to sample like so

• choose random latitudes,
• choose random longitudes
• form the Cartesian product of these lats and longs
• query at those points

This sample is not IID, but can we still say anything about the convergence of the sample mean to the population mean.

Cool question! The samples aren't iid, but you can partition them into subsets that are each internally iid. If you've chosen $$x_1, ... x_n$$ and $$y_1, ..., y_n$$ all independent, then for any fixed $$j$$, you can select a subsequence of iid samples by only using samples of the form $$x_i, y_{i+j}$$ (where the $$y$$ index is modulo $$n$$). The average over each subsequence should converge under the usual assumptions (e.g. no half-Cauchy-distributed mountain summits). The grand mean can then be written as an average of those terms, one per $$j$$, and each of them converges individually, which guarantees convergence overall (by Slutsky's Theorem).
EDIT: For the geographical area example, your $$x$$ and $$y$$ samples should be uniform. In other words, to approximate $$\frac{\int h(x,y)dxdy}{\int dx dy}$$, you rewrite as $$\int h(x,y)f(x)g(x)dxdy$$ where $$f,g$$ are the PDF's you're sampling from. Those PDF's should be constant. Otherwise you're doing importance sampling, and you need to account for the difference between the weights of the average you want (uniform/constant) and the weights of the average you're computing ($$f(x)g(y)$$ where $$f$$ and $$g$$ are the actual pdf's you're using).