# Almost sure convergence of variable

Let $$(X_1, Y_1),...$$ be iid distributed $$\mathbb{R}^2$$ RV's. Assume $$X_i$$ and $$Y_i$$ have first moment $$1$$ and second moment 2. Further assume $$E X_i Y_i = 1-p$$

Let $$T_n = \frac{\sum_{i=1}^n X_i - Y_i}{\sum_{i=1}^n X_i + Y_i}$$

a) Show $$T_n$$ converges almost surely, and identify the limit.

b) Show further that $$T_n$$ has a asymptotic normal distribution, and identify the parameters.

--- My attempt ---

I thought maybe some kind of strong law of large number or CLT could work. However, when checking the conditions, e.g. $$E T_n$$, I get stuck, since $$(X_i - Y_i)$$ and $$(X_i + Y_i)$$ is not independent, therefore, moment computations get hard.. I thought about using the moment assumption about $$X_i Y_i$$, but I cannot seem to get it to work.

I also thought about using some ergodic version of a SLLN. Maybe using a ergodic transformation theorem.

Getting at $$T_n = \phi((X_n,Y_n),...) = \phi \circ S^{n-1}(\mathbb{(X,Y)})$$

where $$\phi((x_1,y_1),...) = \frac{x_i - y_i}{x_i + y_i}$$

I could also just put 1/n into both the numerator and the denominator, and see that every term goes to 1, and therefore $$T_n$$ goes to $$0/2 = 0$$

But then how to do the second part? Just do the same with $$\sqrt{n}$$?

• In your definition of $T_n$, where is $n$? Do you mean $T_n = \sum_{i = 1}^{\color{red}{n}}(X_i - Y_i)/\sum_{i = 1}^{\color{red}{n}}(X_i + Y_i)$? Commented Nov 4, 2022 at 20:12
• In addition, the given conditions do not seem to rule out the possibility that the denominator could be $0$. Commented Nov 4, 2022 at 20:14
• Ahh Yeah of course. It should be n, not infinity. Commented Nov 4, 2022 at 21:07
• Hmm. It is given to me as an exercise. I guess we just have to assume it Canont be 0 Commented Nov 4, 2022 at 21:11
• By SLLN, $n^{-1}\sum_{i = 1}^n(X_i + Y_i) \to E(X_1 + Y_1) = 2$ almost surely so the denominator is non-zero with probability $1$ when $n$ is sufficiently large. So it is a minor concern. Commented Nov 4, 2022 at 21:33

(a) Your attempt is correct. To formalize the argument, you can use something like "if $$a_n \to a, b_n \to b \neq 0$$, then $$a_n/b_n \to a/b$$" in calculus.
(b) Define $$Z_i = (X_i - Y_i, X_i + Y_i)', i = 1, 2, \ldots$$, a sequence of i.i.d. 2D random vectors. Then apply multidimensional CLT and multivariate delta method with respect to $$\{Z_n\}$$.