Suppose we have two sequences of random variables $\{X_n\}$ and $\{Y_n\}$ with $X_n \ge Y_n \ge 1$ such that $\mathbb{E}[X_n - Y_n] < c$ (for some constant $c$).
Is it possible $\mathbb{E}[X_n/Y_n]$ is unbounded?
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Do you aasume $X_n$ and $Y_n$ independent? – Clement C. Sep 11 '19 at 19:01
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Have you tried something? – IamKnull Sep 11 '19 at 19:06
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Note: with the condition $X_n \geq Y_n$, my previous comment makes little sense, but the question becomes much easier. – Clement C. Sep 11 '19 at 19:11
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1$X_n/Y_n - 1 = (X_n -Y_n)/Y_n \le X_n-Y_n$, since $Y_n \ge 1$. – Theoretical Economist Sep 11 '19 at 19:16
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Since $X_n \ge Y_n$, we have that $\frac{X_n}{Yn}\ge 1$. Thus, $$ 0 \le \frac{X_n}{Y_n} - 1 = \frac{X_n -Y_n}{Y_n} \le X_n -Y_n.$$ The last inequality follows from the fact that $Y_n \ge 1$. We therefore have that $$ 0 \le \mathbb E \left[ \frac{X_n}{Y_n}\right] - 1 \le \mathbb E [X_n - Y_n]. $$ Thus, $\mathbb E \left[ X_n/Y_n\right]$ must be bounded.
Theoretical Economist
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