Mathematics – Probability
Scientific paper
2007-12-17
Stochastic Processes and Their Applications, 1221 (2011), 288-313
Mathematics
Probability
Scientific paper
Let $\xi_1,\xi_2,...$ be i.i.d. random variables with negative mean. Suppose that $\mathbf{E}\exp(\lambda\xi_1)<\infty$ for some $\lambda>0$ and that there exists $\gamma>0$ with $\mathbf{E}\exp(\gamma\xi_1)=1$ . It is known that if, in addition, $\mathbf{E} \xi_1\exp(\gamma\xi_1)<\infty$, then the most likely way for the random walk $S_k=\sum_{i=1}^k\xi_i$ to reach a high level is to follow a straight line with a positive slope. We study the case where $\mathbf{E} \xi_1\exp(\gamma\xi_1)=\infty$. Assuming that the distribution $F(dx)=\exp(\gamma x) \mathbf{P}(\xi_1\in dx) $ belongs to the domain of attraction of a spectrally positive stable law, we obtain a weak convergence limit theorem as $r\to\infty$ for the conditional distribution of the process $\bl(r^{-1}\sum_{i=1}^{\lfloor t/ F (r,\infty)\rfloor}\xi_i, t\ge0\br)$ stopped at the time when it reaches level 1 given that the latter event occurs. The limit is an increasing jump process. It is shown to be distributed as an increasing stable L\'evy process stopped at the time when it reaches level 1 conditioned on the event this level is not overshot. Some properties of this process are studied.
Foss Sergey G.
Puhalskii Anatolii A.
No associations
LandOfFree
Random Walks and Levy processes Conditioned not to Overshoot does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Random Walks and Levy processes Conditioned not to Overshoot, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random Walks and Levy processes Conditioned not to Overshoot will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-115820