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The lower envelope of positive self-similar Markov processes. (English) Zbl 1127.60034
Summary: We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and \(+\infty\). Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to A. Dvoretzky and P. Erdős [Proc. Berkeley Sympos. math. Statist. Probability, California July 31–August 12, 1951, 353–367 (1951; Zbl 0044.14001)], M. Motoo [Ann. Inst. Stat. Math. 10, 21–28 (1958; Zbl 0084.35801)], and V. Rivero [Stochastics Stochastics Rep. 75, No. 6, 443–472 (2003; Zbl 1053.60027)].

MSC:
60G18 Self-similar stochastic processes
60G51 Processes with independent increments; Lévy processes
60B10 Convergence of probability measures
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