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 $\sqrt{n}$?

  • $\begingroup$ In your definition of $T_n$, where is $n$? Do you mean $T_n = \sum_{i = 1}^{\color{red}{n}}(X_i - Y_i)/\sum_{i = 1}^{\color{red}{n}}(X_i + Y_i)$? $\endgroup$
    – Zhanxiong
    Commented Nov 4, 2022 at 20:12
  • $\begingroup$ In addition, the given conditions do not seem to rule out the possibility that the denominator could be $0$. $\endgroup$
    – Zhanxiong
    Commented Nov 4, 2022 at 20:14
  • $\begingroup$ Ahh Yeah of course. It should be n, not infinity. $\endgroup$ Commented Nov 4, 2022 at 21:07
  • $\begingroup$ Hmm. It is given to me as an exercise. I guess we just have to assume it Canont be 0 $\endgroup$ Commented Nov 4, 2022 at 21:11
  • $\begingroup$ 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. $\endgroup$
    – Zhanxiong
    Commented Nov 4, 2022 at 21:33

1 Answer 1



(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 $\{Z_n\}$.


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