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Random walks at random times: convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes. (English) Zbl 1306.60038

Authors’ abstract: For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion.
D. Khoshnevisan and T. M. Lewis [Ann. Appl. Probab. 9, No. 3, 629–667 (1999; Zbl 0956.60054)] suggested “the existence of a form of measure-theoretic duality” between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in “alternating” scenery, thus hinting at a mechanism behind this duality.
Following S. Cohen and G. Samorodnitsky [Ann. Appl. Probab. 16, No. 3, 1432–1461 (2006; Zbl 1133.60016)], we also consider alternating random reward schemes associated to random walks at random times. Whereas random reward schemes scale to local time fractional stable motions, we show that the alternating random reward schemes scale to indicator fractional stable motions. Finally, we show that one may recursively “subordinate” random time processes to get new local time and indicator fractional stable motions and new stable processes in random scenery or at random times. When \(\alpha=2\), the fractional stable motions given by the recursion are fractional Brownian motions with dyadic \(H \in (0,1)\). Also, we see that “un-subordinating” via a time-change allows one to, in some sense, extract Brownian motion from fractional Brownian motions with \(H<1/2\).

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G50 Sums of independent random variables; random walks
60G52 Stable stochastic processes
60G51 Processes with independent increments; Lévy processes
60G18 Self-similar stochastic processes
60F05 Central limit and other weak theorems

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