Consider the probability distribution
$$Q(x)=\frac{\beta x}{2}e^{-\frac{\beta}{4}(a^{2}+x^{2})}I_{0}(\frac{\beta a x}{2})$$
where $a$ and $\beta$ are positive numbers and $I_{0}$ is the modified Bessel function of the first kind and order zero.
I'd like to get an estimate (in function of $\epsilon$) for the following integral
$$\int_{\epsilon}^{+\infty}\frac{dx}{Q(x)}[\int_{x}^{+\infty}Q(y)dy]^{2}.$$
I see that, as $Q(x)$ is a probability distribution, this integral should be dominated by the regions where $Q(x)\rightarrow 0$ such as small and large $x$. For small $x$, the integral between brackets is $1$, but then I get a constant $\times$ $\int_{\epsilon}^{+\infty}\frac{e^{\frac{\beta}{4}x^{2}}}{xI_{0}(\frac{\beta a x}{2})}dx\sim\int_{\epsilon}^{+\infty}\frac{1}{x}dx$, integral that diverges (while the original one I'm pretty confident converges).
I thought about dividing this integral in $[\epsilon,R]$ and then $[R,+\infty]$ (that way the logarithm dependence from $1/x$ is constrained in the $[\epsilon,R]$ interval), but the other integral becomes very cumbersome and I think I can only approximate it for $R\rightarrow 0$.
There has to be a way of estimating the value of this integral through asymptotic analysis (tried with the Watson's lemma around zero but that led nowhere). I don't necessarily need a closed-form solution, but at least an approximation that works well, maybe for small values of $\epsilon$. Any suggestions are appreciated!
Best Answer
For large positive $z$, $$ I_0 (z) \sim \frac{{e^z }}{{\sqrt {2\pi z} }}. $$ Thus $$ Q(y) \sim \frac{1}{2}\sqrt {\frac{\beta }{{\pi a}}} e^{ - \beta a^2 /4} \sqrt y e^{ - \beta y^2 /4 + \beta ay/2} $$ for large $y$ with positive $a$ and $\beta$. By L'Hôpital's rule, it follows that $$ \int_x^{ + \infty } {\sqrt y e^{ - \beta y^2 /4 + \beta ay/2} dy} \sim \frac{2}{{\beta \sqrt x }}e^{ - \beta x^2 /4 + \beta ax/2} $$ for large $x$. Finally, $$ \frac{1}{{Q(x)}}\left( {\int_x^{ + \infty } {Q(y)dy} } \right)^2 \sim \frac{2}{{\beta ^{3/2} \sqrt {\pi a} }}e^{ - \beta a^2 /4} \frac{1}{{x^{3/2} }}e^{ - \beta x^2 /4 + \beta ax/2} $$ for large $x$, which is integrable. I would write your integral as $$ \int_\varepsilon ^X {\frac{1}{{Q(x)}}\left( {\int_x^X {Q(y)dy} + \int_X^{ + \infty } {Q(y)dy} } \right)^2 dx} + \int_X^{ + \infty } {\frac{1}{{Q(x)}}\left( {\int_x^{ + \infty } {Q(y)dy} } \right)^2 dx} $$ with a suitably large $X>0$. The integrals along $(X,+\infty)$ can be estimated by the asymptotic formulae I provided above. For the integrals over the finite intervals, you may use convergent power series to estimate $Q$.