Invariance principles for random walks conditioned to stay positive

Caravenna, Francesco; Chaumont, Loïc

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 170-190.

Soit $\left\{S_n\right\}$ une marche aléatoire dont la loi est dans le domaine d’attraction d’une loi stable $𝒴$, i.e. il existe une suite de réels positifs $\left(a_n\right)$ telle que $S_n/a_n$ converge en loi vers $𝒴$. Nous montrons que le processus renormalisé $(S_{⌊nt⌋}/a_n,t\ge 0)$, une fois conditionné à rester positif, converge en loi (au sens fonctionnel) vers le processus de Lévy stable de loi $𝒴$ conditionné à rester positif. Sous certaines hypothèses supplémentaires, nous montrons un principe d’invariance pour cette marche aléatoire tuée lorsqu’elle quitte la demi-droite positive et conditionnée à mourir en 0.

Let $\left\{S_n\right\}$ be a random walk in the domain of attraction of a stable law $𝒴$, i.e. there exists a sequence of positive real numbers $\left(a_n\right)$ such that $S_n/a_n$ converges in law to $𝒴$. Our main result is that the rescaled process $(S_{⌊nt⌋}/a_n,t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy 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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