Probability Limits

Question

What is the probability limit of $\frac{\frac{1}{n}{\sum}_{i = 1}^{n} X_i}{\frac{1}{n}{\sum}_{i = 1}^{n} Y_i}$ where $X_1$, ..., $X_n$, $Y_1$, ...$Y_n$ both come from a distribution with mean $\mu$ and variance $\sigma^2$

I know that the probability limit of $\frac{1}{n}{\sum}_{i = 1}^{n} X_i$ is just $\mu$ from the law of large numbers, but how do you solve for this when its these two fractions on top of each other? Does a probability limit even exist?

Thank you

Answer 1

There I will assume, that $X_1, ... X_n, Y_1, ... Y_n$ are i.i.d.

In this case, your problem splits in two cases:

1. $\mu \neq 0$

Then $\frac{\sum_{i = 1}^n X_i}{\sum_{i = 1}^n Y_i}$ converges almost surely to $1$ by the Strong Law of Large Numbers.

2. $\mu = 0$

Then we have by CLT, that $\frac{\sum_{i = 1}^n X_i}{\sigma \sqrt{n}}$ converges in distribution to $N(0, 1)$. The same goes for $\frac{\sum_{i = 1}^n Y_i}{\sigma \sqrt{n}}$. And because they are independent, we have that $\frac{\sum_{i = 1}^n X_i}{\sum_{i = 1}^n Y_i}$ converges in distribution to $Cauchy(0, 1)$