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Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively. Assume both $X$ and $Y$ take strictly positive values.

Consider the random variable $R_n \doteq \frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$.

I am searching for a generalization of the notion that if $X$ and $Y$ have finite mean and variance, $R_n$ should approach $\frac{\mathbb{E}X}{\mathbb{E}Y}$ an $n \to \infty$.

In the more general case when $X$ and $Y$ may not have finite expectations, can some notion of concentration be used to argue that $R_n$ approaches some simpler distribution (which may be a function of $n$)?

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  • $\begingroup$ What if $\mathbb{E} X = \mathbb{E} Y = 0$? $\endgroup$ – usul Dec 3 '15 at 16:36
  • $\begingroup$ I'm interested in the case where both are strictly positive, so I will make an edit to that effect. $\endgroup$ – Patrick Sanan Dec 3 '15 at 21:50
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    $\begingroup$ can't you just use the generalized limit theorem, so that $R_n$ approaches the ratio of two independent variables with a Cauchy or Lévy distribution (depending on whether mean or variance are finite) $\endgroup$ – Carlo Beenakker Dec 3 '15 at 22:00
  • $\begingroup$ What sort of result do you want if $Y$ is constant while $\mathbb{E}X=\infty$? $\endgroup$ – Douglas Zare Dec 3 '15 at 22:55
  • $\begingroup$ @CarloBeenakker, that does seem like the correct approach. $\endgroup$ – Patrick Sanan Dec 5 '15 at 14:45
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This isn't a real answer: just an observation of some nasty examples with infinite mean.

First off, you can take $X$ to have probability $2^{-m}$ on $T(2m)$ for each $m\ge 1$, where $T$ is the tower function (or any other very fast-growing function), and $Y$ to have weight $2^{-m}$ on $T(2m+1)$, and both probability $0$ elsewhere. If you do this, the asymptotic value of each sum is more or less given by the single largest value in either sum, so one of the two sums is much larger than the other. So the distribution of the limit ratio has all the weight at $0$ and $\infty$ (taken as a compactification point).

Another similar example: both $X$ and $Y$ have probability $2^{-m}$ on $T(m)$ and $0$ elsewhere. Now the sum in the numerator is basically an integer multiple of the largest individual term, same for the denominator. So the limit distribution is the ratio of two random variables whose distribution is Poisson with some extra weight thrown on to $0$ and $\infty$, so the distribution in the limit has positive weight on the rationals and is zero elsewhere.

You can play further with this kind of example to have all kinds of oscillatory or generally nasty behaviour as $n$ grows. For some range of $m$, take the first example, for larger $m$ the second, return to the first, and so on. For large $n$ you only see one of the two regimes, so the distribution oscillates from something that looks like it tends to a two-point limit to the horrible second limit and back.

So as soon as you allow arbitrary $X$ and $Y$, you can have some pretty wild behaviour in the limit.

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