Unimodality of hitting times for stable processes

Julien Letemplier (1) , Thomas Simon (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

We show that the hitting times for points of real $\alpha-$stable Lévy processes ($1<\alpha\le 2$) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al., whereas in the completely asymmetric case we apply an identity of the second author. The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al.

Keywords: Hitting time; Kanter random variable; Self-decomposability; Size bias; Stable process; Unimodality

Domains: Probability [math.PR]

Julien Letemplier, Thomas Simon. Unimodality of hitting times for stable processes. 2013. ⟨hal-00864286v1⟩