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Published in:
2021 | OriginalPaper | Chapter
Two Continua of Embedded Regenerative Sets
Authors : Steven N. Evans, Mehdi Ouaki
Published in: A Lifetime of Excursions Through Random Walks and Lévy Processes
Publisher: Springer International Publishing
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Abstract
Given a two-sided real-valued Lévy process \((X_t)_{t \in \mathbb {R}}\), define processes \((L_t)_{t \in \mathbb {R}}\) and \((M_t)_{t \in \mathbb {R}}\) by \(L_t := \sup \{h \in \mathbb {R} : h - \alpha (t-s) \le X_s \text{ for all } s \le t\} = \inf \{X_s + \alpha (t-s) : s \le t\}\), \(t \in \mathbb {R}\), and \(M_t := \sup \{ h \in \mathbb {R} : h - \alpha |t-s| \leq X_s \text{ for all } s \in \mathbb {R} \} = \inf \{X_s + \alpha |t-s| : s \in \mathbb {R}\}\), \(t \in \mathbb {R}\). The corresponding contact sets are the random sets \(\mathcal {H}_\alpha := \{ t \in \mathbb {R} : X_{t}\wedge X_{t-} = L_t\}\) and \(\mathcal {Z}_\alpha := \{ t \in \mathbb {R} : X_{t}\wedge X_{t-} = M_t\}\). For a fixed \(\alpha >\mathbb {E}[X_1]\) (resp. \(\alpha >|\mathbb {E}[X_1]|\)) the set \(\mathcal {H}_\alpha \) (resp. \(\mathcal {Z}_\alpha \)) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections \((\mathcal {H}_{\alpha })_{\alpha > \mathbb {E}[X_1]}\) and \((\mathcal {Z}_{\alpha })_{\alpha > |\mathbb {E}[X_1]|}\) are increasing in α and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that \((\sup \{t < 0 : t \in \mathcal {H}_\alpha \})_{\alpha > \mathbb {E}[X_1]}\) is a càdlàg, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for \((\sup \{t < 0 : t \in \mathcal {Z}_\alpha \})_{\alpha > |\beta |}\) when \((X_t)_{t \in \mathbb {R}}\) is a (two-sided) Brownian motion with drift β.