Let $Y_i$, $X_i$ be sequences of independent random variables. Assume both limits exist: $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} VarX_i}{\sum_{i=1}^{n} VarY_i}, \lim_{n \to \infty} \frac{\sum_{i=1}^{n} \mathbb{E}X_i}{\sum_{i=1}^{n} \mathbb{E} Y_i}$$.

Does it imply this limit exists : $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} X_i}{\sum_{i=1}^{n} Y_i}$$

Any ideas whether it's true or not? The limit converging almost surely that is. Maybe Kolmogorov's two series theorem could help?