Deterministic time intervals on which a class of persistent processes are away from their origins. (English) Zbl 1348.60052

Summary: There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur. The first result gives a sufficient condition for a persistent random walk with a finite fourth moment. In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin. The second result gives a necessary and a sufficient condition for similar excursions in the case of Brownian motion. The third result gives a necessary and a sufficient condition for time intervals to be free of the zeros of a class of persistent natural scale linear diffusions on the line and is equivalent to the determination of recurrent sets at infinity of the inverse local time.


60F20 Zero-one laws
60G50 Sums of independent random variables; random walks
60J60 Diffusion processes
60J65 Brownian motion
60J55 Local time and additive functionals
60K99 Special processes
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