Return to Question

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?

Might be worth to note, the series in the numerator/denominator alone may not converge.

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

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?

Might be worth to note, the series in the numerator/denominator alone may not converge.

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?

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?

Might be worth to note, the series in the numerator/denominator alone may not converge.

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?

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?