Timeline for Some Cramér-Lundberg asymptotics analogue for this simpler compound Poisson process?
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| Aug 21, 2017 at 9:59 | comment | added | Alexandre | Ok, that's a really nice insight! Then, the first moment of the cdf $S(a)$ in bounded $\langle A \rangle\leq \nu^{-1}$, and it remains to figure out if we can get a sharper estimate. | |
| Aug 21, 2017 at 8:57 | comment | added | Mateusz Kwaśnicki | Well, it does tell you more. If you look into the proof of the result that you linked to, you will realise that $e^{\nu x}\mathbf{P}(\overline{X}_\infty<x)=\mathbf{P}(\overline{X}_\infty^{(\nu)}<\infty)$ for a Lévy process $X_t^{(\nu)}$ obtained by an exponential change of measure. This implies that if $X_t$ is as in your question, then $e^{\nu a}(1-S(a))$ is the tail of survival probability of a different compound Poisson process (with jumps distributed as $ce^{-\nu x}f(x)$ for appropriate $\nu$ and $c$). In particular, $1-S(a)\leqslant e^{-\nu a}$. | |
| Aug 20, 2017 at 23:52 | comment | added | Alexandre | Ok, reading about fluctuation of Lévy processes, I found a theorem: Cramér's estimate (p.207), which gives the conditions for the exponential behaviour of the tail of the survival probability. This is sufficient to say that its moments are finite, but we cannot say much more. For the moment, I can not find a generalization of the Lundberg inequality which states the conditions such that the whole survival probability is bounded by above by the exponential distribution (with the same exponent as the tail). | |
| Aug 20, 2017 at 20:15 | history | edited | Alexandre | CC BY-SA 3.0 |
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| Aug 20, 2017 at 19:54 | comment | added | Alexandre | Thank you for these references. Yes I am interested in the case of a positive bias, since the survival probability is zero otherwise, and I expect a distribution which is asymptotically exponential. I will add some details. | |
| Aug 20, 2017 at 19:32 | comment | added | Mateusz Kwaśnicki | (second part of the above comment) In this case the asymptotic behaviour of the survival probability depends on the properties of the distribution of $X_n$, and in order to tell something more, it would be convenient to know what do we assume about (the tail of) $X_1$. | |
| Aug 20, 2017 at 19:29 | comment | added | Mateusz Kwaśnicki | This is the basic question in fluctuation theory of Lévy processes, which provides you with a lot of tools to study it. For compound Poisson processes, the "discrete" variant for random walks is typically sufficient, and a standard reference here is Feller. For processes in continuous time, see either Bertoin's book on Lévy processes, or any book that combines "fluctuation" and "Lévy" in its title. In principle, if $\langle X_n\rangle>0$, then the survival probability has a positive limit at infinity, so I suppose you meant $\langle X_n\rangle<0$. (to be continued) | |
| Aug 20, 2017 at 17:31 | history | edited | Alexandre | CC BY-SA 3.0 |
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| Aug 20, 2017 at 17:15 | history | asked | Alexandre | CC BY-SA 3.0 |