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For a given $\alpha \in (0,2)$ How fast does

\begin{equation} \int_{\pi/h}^\infty{\exp(-p^\alpha)}\,\mathrm{d}p \end{equation}

go to zero as $h$ goes to zero? Any upper bound on the speed of convergence would be great!

1 Answers 1

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Hint: use integration by parts

Hint2: to simplify the calculation. Substitute first $x=p^\alpha$

Hint3: you should get the result $\sim (\pi/h)^{1-\alpha} e^{-(\pi/h)^\alpha} \alpha^{-1}$

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    Also , I am not so sure that asymptotic is true anymore since: $\int_{(\frac{\pi}{h})}^\infty e^{-p^\alpha}dp = \alpha^{-1}(\frac{\pi}{h})^{1-\alpha}e^{-{\frac{\pi}{h}}^\alpha} + \epsilon(h)$ where $\epsilon(h) = o(1)$ as $h$ goes to zero and is given by $\int_{{\frac{\pi}{h}}^\alpha}^\infty \frac{1}{\alpha}(\frac{1}{\alpha} -1) x^{\frac{1}{\alpha}-2}e^{-x}dx$ which may go to zero slower than the first term ( all we know so far is that is o(1)). Do you have any comments on that?thanks2012-08-07