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Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. (English) Zbl 1098.60038

A càdlàg Markov process with family of probability laws \((\mathbb{P}_x)_{x \geq 0}\) is called self-similar if it fulfils a time-space-scaling property, that is, there exists a constant \(\alpha>0\) such that the law of \((kX_{k^{-\alpha}t})_{t \geq 0}\) under \(\mathbb{P}_x\) is given by \(\mathbb{P}_{kx}\) for all \(k>0\). The authors focus on positive (i.e. \(\mathbb{R}^+\)-valued) self-similar Markov processes. Well known examples as Bessel processes, stable subordinators, or more generally, stable Lévy processes conditioned to stay positive, are discussed.
The self-similarity implies weak continuity of the family \((\mathbb{P}_x)\) in \(x\) on the open half line \((0, \infty)\). The authors answer the question of the existence of a weak limit when \(x\) goes to \(0\). More precisely, they first give conditions which allow to construct a càdlàg Markov process \(X^{(0)}\), starting from \(0\), which stays positive and possesses the scaling property. Secondly, they establish necessary and sufficient conditions for the laws \((\mathbb{P}_x)\) to converge weakly to the law of \(X^{(0)}\) as \(x \to 0\). In doing so, a crucial point is the Lamperti representation of self-similar \(\mathbb{R}^+\)-valued processes, which provides the construction of the paths of any such self-similar process \(X\) from those of a Lévy process [see J. Lamperti, Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 205–225 (1972; Zbl 0274.60052)].

MSC:

60G18 Self-similar stochastic processes
60J25 Continuous-time Markov processes on general state spaces
60B10 Convergence of probability measures
60G51 Processes with independent increments; Lévy processes

Citations:

Zbl 0274.60052
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References:

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