Stochastic Taylor series

Hi,

I am wondering if the following type of Taylor series is known .

I am working with the following quantity:
z = ArcTan[ (sin(a) + x)/(cos(a) + y) ]
where x,y i.i.d. N(0,s^2] where s<1 (we'll say s<<1 to simplify the
problem)

I want to study the statistical properties of z. Now, the distribution
of z can be worked out (since the distribution of the ratio of two
independent normal rvs is a known one), but in my case, the
distribution is too complicated to work with very easily.

As a consequence, I want to do a Taylor series expansion of z wrt x
and y (about x=0 and y=0).
The form for the Taylor series has a nice simple form (esp for the
first few orders):
z = a + cos(a)* x - sin(a)* y - sin(2a) *(x^2/2) - sin(2a)*(y^2/2) -
cos(2a) * x*y.

Since x and y are statistically independent, computing approximations
to the mean, variance, etc of z can then be performed rather easily.

Technically, however, I am wondering how valid this is. x and y will
almost surely have a magnitude less than unity, but cases may arise in
which the magnitude of x relative to cos(a) may not be small, and it
is not obvious to me that that wouldn't cause problems.

Thanks in advance for any help/insight on whether people perform this
type of Taylor series expansions and what conditions are required for
them to be well-defined.

Peter
.

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