🚀 go-pugleaf

Hi, there, could someone out there help me with the following
ratio estimate problem ? Any comments and/or references are
appreciated. Now the problem:

Suppose {X(i)} and {Y(i)} are i.i.d. sequences respectively, but
X(i) and Y(i) may be correlated. Let m(x) and m(y) be the expected
values of X(i) and Y(i) repectively. We would like to estimate 
	    m(x)
	R = ----. 	We know that
	    m(y)


	^      [ sum_1_to_n X(i) ] / n
	R(n) = -----------------------  
               [ sum_1_to_n Y(i) ] / n

is a consistent but biased estimator for R. Does anyone know of
any RIGOROUS treatment on the variance of this estimator ?
Particularly, I am interested in some sufficient conditions of X(i),Y(i),
that imply
                           ^
lim_n_to_infinity   n var( R(n) ) 

         E(X**2)    E(Y**2)      E(X*Y)
= R**2 { -------  + ------- - 2--------- }.
         m(x)**2    m(y)**2    m(x)*m(y)


Comments on more general nonlinear estimator are also welcome.

Thanks a lot.

--J. Wang