Let us consider a real-valued r.v. $X$ such that $\mathsf E X > 0$ and $X>x_0$ a.s. Denote $ g(x,y) = \frac{x}{y(y+x)}. $ It is right that for any $X$ there exists $y_0>-x_0$ such that for all $y>y_0$ one have $\mathsf E g(X,y)>0$?
Estimation of an expectation
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probability
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0Thanks for noting this. (I deleted my second comment.) – 2011-06-21
1 Answers
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For every $x$ and $z$ such that $1+zx$ is positive, let $h(x,z)=x/(1+zx)$. For every positive $z$ such that $z\le1/(2|x_0|)$, one has $1+zx_0\ge\frac12$, hence $|h(X,z)|\le2|X|$. Furthermore, $h(X,z)\to X$ when $z\to0^+$. By Lebesgue dominated convergence theorem, $E(h(X,z))\to E(X)$ when $z\to 0^+$.
Since $E(X)$ is positive, this implies that there exists a positive $z_0$ such that for every $z$ in $(0,z_0)$, $E(h(X,z))\ge E(X)-\frac12E(X)$ hence $E(h(X,z))$ is positive.
Note finally that $g(x,y)=h(x,1/y)/y^2$ hence for every positive $y$, $E(g(X,y))$ is positive if and only if $E(h(X,1/y))$ is. This happens at least for every $y\ge1/z_0$.
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0Thank you, that was helpful for the question of bounds for submartingales. – 2011-06-20