Invariance principles for random walks conditioned to stay positive

Details Invariance principles for random walks conditioned to stay positive Invariance principles for random walks conditioned to stay positive

2006-02-14

arxiv.org/abs/math/0602306v2

Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 1, 170-190

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AIHP119 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu

Scientific paper

10.1214/07-AIHP119

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable L\'{e}vy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

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