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I've been fighting with this homework problem for a while now, and I can't quite see the light. The problem is as follows,

Assume random variable $X \ge 0$, but do NOT assume that $\mathbb{E}\left[\frac1{X}\right] < \infty$. Show that $\lim_{y \to 0^+}\left(y \, \mathbb{E}\left[\frac{1}{X} ; X > y\right]\right) = 0$

After some thinking, I've found that I can bound

$ \mathbb{E}[1/X;X>y] = \int_y^{\infty}\frac1{x}\mathrm dP(x) \le \int_y^{\infty}\frac1{y}\mathrm dP(x) $

since $\frac1{y} = \sup\limits_{x \in (y, \infty)} \frac1{x}$ resulting in

$ \lim_{y \to 0^+} y \mathbb{E}[1/X; X>y] \le \lim_{y \to 0^+} y \int_y^{\infty}\frac1{y}\mathrm dP(x) = P[X>0]\le1 $

Of course, $1 \not= 0$. I'm not really sure how to proceed...

EDIT: $\mathbb{E}[1/X;X>y]$ is defined to be $\int_y^{\infty} \frac{1}{x}\mathrm dP(x)$. This is the notation used in Durret's Probability: Theory and Examples. It is NOT a conditional expectation, but rather a specifier of what set is being integrated over.

EDIT: Changed $\lim_{y \rightarrow 0^-}$ to $\lim_{y \rightarrow 0^+}$; this was a typo.

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    @Sri yup, typo. Thanks for catching!2011-09-06

2 Answers 2

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Hint(s) For every positive $y$, let $Z_y=(y/X)\mathbf{1}(y/X<1)$. You want to prove that $E(Z_y)\to0$ when $y\to0^+$.

It happens that $0\le Z_y<1$ with full probability, for every positive $y$, and that $Z_y\to0$ when $y\to0^+$ with full probability. (Of course you should check this.)

Now, your goal is to find a condition on a given family of nonnegative random variables $(T_y)$ that ensures that $\lim\limits_{y\to0^+}E(T_y)=E\left(\lim\limits_{y\to0^+}T_y\right)$ and to check that your family $(Z_y)$ fulfills this condition. There should not exist so many conditions of this ilk in your textbook...

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    @Sasha, either bounded or dominated convergence should work here.2011-09-21
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Hint: For any $k > 1$, $\int_y^\infty \frac{y}{x} \ dP(x) \le \int_y^{ky}\ dP(x) + \int_{ky}^\infty \frac{1}{k} \ dP(x) \le \ldots$

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    Another way is to make the split at $\sqrt{y}$ instead of $ky$.2011-09-06