Testing for product of Gaussians

Hi

I am posting to sci.math as my original message (under a different
title) in sci.stat.math was ignored - the two stat newsgroups don't
seem as busy as this one. Sorry if this is considered off topic over
here.

I'd like to know an appropriate statistical test for finding out
whether data in 2 dimensions comes from the product of two uncorrelated
normal distributions. To be more specific for a series of (x,y) pairs
I have a value (let's say z) created by a computer simulation. I know
the actual values of z should be drawn from the product of two
independent normal distributions with mean zero and the same standard
deviation in both directions. I wonder if anyone could tell me an
appropriate statistical test to check that computed values have indeed
probably come from this underlying distribution? Given that so many
parametric tests seem to require normal data to work properly, I'd
guess this is a very standard question?

I have a reasonable mathematical background (although sadly I don't
know very much statistics), so a reference to why the test works would
be fantastic too. Looking on the internet I came up with a variety of
possibilities, but thought it better to ask someone who knows before
blindly applying a big formula!

(In case it is relevant or helps explain what the statistical test I
require more exactly, I am attempting to test a numerical solution the
simple spatially homogeneous diffusion equation in 2 dimensions with
delta function initial data. The numerical solution proceeds by
creating a large number of individual particles at the origin and
simulating each one randomly moving in space for a large number of time
steps, with an appropriate step length based on the time step and the
diffusion coefficient. Then at the end of the simulation the program
counts the number of particles in each square on the plane for squares
with small side lengths. I know the answer to the diffusion equation,
so given the number of particles originally released can trivially find
out how many particles I would expect to be in each square given its
position. The simulation appears to give the correct answer, inasmuch
as I get a nice bell shaped surface, but I'd like to be statistically
sure. Before anyone numerically minded raises any objections, I know
this is a terribly inefficient way of solving a diffusion equation, but
I am later going to have to solve a reaction advection diffusion
equation in some horrible geometry with strongly spatially varying
advection so this Lagrangian style approach is a contender)

Many thanks in advance to anyone who can help

.

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