0
$\begingroup$

Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.

Let $\tau_n$ be the stopping time corresponding to the first time $t$ when either $L(t) \ge n$ or $L(t) \le -n$.

I'm interested in the distribution $X_n := [L(\tau_n) - n | L(\tau_n) \ge n]$, the amount by which $L$ jumps over the upper limit $n$. I call this the "exceedance distribution", I don't know if it has a name in the literature.

Question: In the $n \to \infty$ limit, does $X_n$ converge to a limiting distribution? If so, what distribution?

In simulation, $X_n$ appears to converge to the following distribution:

Simulation of exceedance

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

First. if you were only interested in the one-sided problem, then you look in at the jump times and you have the same problem for a random walk which is the difference of a uniform and an exponential. Here you are in the realm of standard renewal theory, the limit absolutely exists and I think you can get it pretty explicitly. Then, to deal with the complication of the 2 sided problem, note that the excess under the lower boundary is always 0, and it would not make any difference if you used the excess or the quantity you gave. If you stop the process when it crosses (n-k, k-n) for some k which is large but small to n, then with high probability the ones that cross k-n also cross -n etc., and cross with their limiting distribution, since k is large. You get a half and half mixture of 0 and the result for the upper boundary. If you are looking for a reference on the renewal theory, I like feller vol 2.

Improve this answer
$\endgroup$
3
  • $\begingroup$ In standard renewal theory, jumps must be nonnegative. Briefly looking at Feller, it seems the same assumption is made there. Do you know of a reference for the generalization of renewal theory to allow negative renewal times? $\endgroup$
    – isaacg
    Commented Feb 7, 2023 at 16:16
  • $\begingroup$ No, much of feller's section is concerned with removing this assumption. The (kind of obvious) solution is to look at and imbedded process with positive increments, the ladder height process. What isn't obvious is how much info you can get about it. Maybe see if Feller uses the term ladder height ? Also, if memory serves, it turns out that the situation that arises in a compound poisson process, where the negative part is exponential, is particularly simple/ $\endgroup$
    – mike
    Commented Feb 8, 2023 at 5:59
  • $\begingroup$ Thanks, I think this puts me on the right path $\endgroup$
    – isaacg
    Commented Feb 8, 2023 at 8:30

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .