Consider the sequence of rectangles, which sides are length $(X_1, Y_1), (X_2, Y_2),...,$ where $X_1, X_2,...$ have uniform distribution on $(0,2)$ and $Y_1, Y_2, ...$ have uniform distributions on $(0,4)$. All random variables are independent, Let $P_1, P_2,...$ be sequence of surfaces of these rectangles and $O_1, O_2,...$ sequence of perimeters of the rectangles. What is the limit of $$ \frac{\sum_{i=1}^{n}{P_i}}{\sum_{i=1}^{n}{O_i}},$$ when $n \rightarrow \infty$?
Doesn't matter what kind of convergace of random variable we consider.
I saw that $P_i$ and $O_i$ are independent, when $i \neq j$.
I tried to put on the ratio $log(x)$ and inspect characteristic function and then use Lévy's continuity theorem [1]. I thought about CLT and law of large numbers.
Please, help! Maybe some advises or intuitions. Thanks!
[1] https://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem