Random Walks and Levy processes Conditioned not to Overshoot

Details Random Walks and Levy processes Conditioned not to Overshoot Random Walks and Levy processes Conditioned not to Overshoot

2007-12-17

arxiv.org/abs/0712.2637v2

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.

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