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Patrick Sanan
Patrick Sanan
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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 $n$)?

Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively.

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 $n$)?

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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Patrick Sanan
Patrick Sanan
  • 143
  • 4

Asymptotic behavior of a ratio of sums of iid random variables

Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively.

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 $n$)?