0
$\begingroup$

If $F(s)$ is the Laplace transform of $f(t)$ and \begin{equation} F(s)=\frac{1}{1-aG(s)} \end{equation} where $G(s)$ is the Laplace transform of a known probability density distribution $g(t)$ whose moments are $\langle t^n\rangle$.

Is it possible to approximate $f(t)$ in terms of the first moments $\langle t^n\rangle$? At least in the limit of large $t\gg1$?

$\endgroup$
1

1 Answer 1

0
$\begingroup$

The large-$t$ behavior of $f(t)$ will be dominated by the small-$s$ behavior of $F(s)$, so it makes sense to include only the first few moments of $g(t)$. If I take a normal distribution for $g$, I obtain $$f(t)=-\frac{\sqrt{2} \sinh (t\sqrt{2/a-2 } )}{\sqrt{a(1-a)}}.$$ So the large-$t$ behavior is either exponential or oscillatory, depending on whether $a$ is larger or smaller than 1.

More generally, for $\langle t\rangle=\mu$, $\langle t^2\rangle=\mu^2+\sigma^2$, one has $$f(t)=-2 e^{\frac{{\mu} t}{{\mu}^2+{\sigma}^2}} \sinh \left(\frac{t \sqrt{(2-a) a {\mu}^2+2 (1-a) a {\sigma}^2}}{a \left({\mu}^2+{\sigma}^2\right)}\right)\biggl((2-a) a {\mu}^2+2 (1-a) a {\sigma}^2\biggr)^{-1/2}.$$
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.