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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.

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    $\begingroup$ "Almost surely" - definitely not: Just take $X_i$ and $Y_i$ independent Gaussians with mean zero (shift the expectations a tiny bit to have the second ratio well-defined if $0/0$ bothers you) and the same variance. Then the ratios for very different $n$ are almost independent but the distribution is all the way the same (that of the ratio of 2 standard Gaussians) and not that of a constant. $\endgroup$
    – fedja
    Commented Sep 5, 2020 at 21:44
  • $\begingroup$ A simple counterexample: $X_i=Y_i=N(1,3^i)$ $\endgroup$
    – user44143
    Commented Sep 6, 2020 at 4:32

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