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On the future infimum of positive self-similar Markov processes. (English) Zbl 1100.60018
An \(\mathbb{R}_{+}\)-valued Markov process \(X=(X_{t})_{t\geq 0}\) with càdlàg paths is said to be a (positive) self-similar process (PSSMP for short) if for every \(k>0\) and every initial state \(x\geq 0,\) and for some \(\alpha >0,\) the law of \((kX_{tk^{-\alpha }})_{t\geq 0}\) under \(\mathbb{P}_x\) is \(\mathbb{P}_{kx}\), where \(\mathbb{P}_x\) is the law of \(X^{(x)}\) (\(X\) started from \(x\geq 0\)). If \(X^{(x)}\) drifts toward \(+\infty\), its future infimum \( J^{(x)}=(J_t^{(x)})_{t\geq 0}\) is defined by \( J_t^{(x)}=\inf_{s\geq t}X_s^{(x)},\;t\geq 0\). The author establishes integral tests and laws of the iterated logarithm for the upper envelope at \(0\) and \(+\infty\) of the future infimum \(J^{(x)}\) and for increasing self-similar Markov processes. The proofs are based on the Lamperti representation of PSSMPs and time reversal arguments due to L. Chaumont and the author. The results obtained extend laws of the iterated logarithm for future infima of Bessel processes derived by D. Khoshnevisan, T. M. Lewis and W. V. Li [Probab. Theory Relat. Fields 99, No. 3, 337–360 (1994; Zbl 0801.60066)].

60G18 Self-similar stochastic processes
60F15 Strong limit theorems
60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes
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