# Derive the asymptotic distribution

Let $$x_1, ..., x_n$$ and $$y_1, ..., y_n$$ be two independent random samples from $$X$$ and $$Y$$. We have $$µ_X = E (X ) > 0, µ_Y = E (Y ) > 0$$ and $$σ^2_X = Var (X )$$ and $$σ^2_Y = Var (Y )$$.

Derive the asymptotic distribution of $$\frac{\overline x_n+ \overline y_n}{\overline x_n- \overline y_n}$$.

where $${\overline x_𝑛}$$ is the sample average of the 𝑥𝑖

Haven't put any additional information because I am hitting a wall, really don't know how to resolve this. I have looked at the delta method as a possible route but it's too convoluted for me

• What have you done so far? Can you edit the question and show your current work? – Brennan Nov 17 '19 at 18:22
• Wished I could tell you more, but I really stuck. Don't really know how to proceed. I know people would rather see me find the solution myself but if I knew how I would have put it down here. Thanks for caring Brennan – James Nov 17 '19 at 18:46

## 1 Answer

I can provide you with a sketch of proof to proceed.

The quantity $$\sqrt{n}(\bar{x}-\mathrm{E}(X)) \overset{d}{\to} N(0,\,\mathrm{V}(X))$$ by Central Limit Theorem. Similarly, the same can be said for the other quantity, $$\bar{y}$$.

Since a linear combination of Gaussian variables are Gaussian, then the summation and subtraction of the two quantities are also Gaussian. Prove that $$\bar{x}+\bar{y}$$ and $$\bar{x}-\bar{y}$$ are of zero means, then the quotient/ratio of two zero-mean Gaussian random variable is a Cauchy distribution.