Deriving the sampling (or asymptotic) distribution

Assume we have the following function:

where

is a constant

is a constant

for are random variables

for are random variables

is defined such that it is value that satisfies

I wish to derive a sampling (or asymptotic) distribution for the statistic .

By sampling distribution I mean the following:

The solution to doesn't have a closed-form solution, but it is obvious that the resulting value of depends on and , so can be treated as a random variable that depends on the random variables and . Then for every observations of and , we have a corresponding value that satisfies , what is the sampling distribution of ?

By asymptotic distribution I mean the following:

Assume we have instances of and , that is, groups of and . Then we solve and have observations of , that is, . What is the distribution of as ?

Also assume you are allowed the following assumptions to achieve the above:

1) You can make any distributional assumptions regarding and , e.g., and are independent from each other, also , for are independently and identically distributed.

2) Rather than making distributional assumptions about and , assume you can make some assumptions about the processes and , e.g., both processes are stationary (or weakly stationary) etc.

Re: Deriving the sampling (or asymptotic) distribution

Usually when you are talking about the asymptotics of a certain statistic, it should be a sequence indexed by the sample size . However, now it seems that the group you defined are independent to each other so that are just a sequence of i.i.d. random variables which does not related to any asymptotic result. Am I missing anything?

Re: Deriving the sampling (or asymptotic) distribution

You are quite right. Yes so the relation to which you have simplified to is the one I am after. Assume we fix , how can we derive the distribution of ?