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I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability distribution and $\mu$ is some complex variable with strictly positive real part $\mu_r>0$.

It is easy enough to prove that this function is bounded above, $|f(\mu)| \le \frac{1}{\mu_r}$.

However, numerics suggest that for 'nice' distributions (symmetric, and non-increasing on $[0,\infty)$), the integral is bounded below by $|f(\mu_r)|\le|f(\mu)|$.

So my questions are:

1) Are there established results on finding lower bounds of the Laplace transform of a characteristic function. (My trawling of google scholar and the like haven't produced anything. It seems like such an elementary problem that surely someone has considered it before.)

2) If not what general techniques are used to look for lower bounds on integrals such as this?

Background: The integral comes from investigating how non-identical frequency distribution of linear oscillators a particular collective behavior problem. This has occurred before in such problems and has been treated asymptotically for strictly real $\mu$ (ref. 1) and commented on for complex $\mu$ (appendix ref. 2). Due to the parameter regions we are interested in (around $\Re{\mu} =1$) the approximations made previously no longer hold.

Many Thanks, Pete.


Ref. 1 - RE. Mirollo, SH Strogatz. Amplitude Death in Limit Cycle Oscillators. http://www.springerlink.com/index/ln051rp502550471.pdf

Ref. 2 - PC. Matthews, RE. Mirollo, SH Strogatz. Dynamics of a large system of coupled nonlinear oscillators. http://www.sciencedirect.com/science/article/pii/016727899190129W

edit: (typo) corrected to non-increasing

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