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July 2013 Random walks at random times: Convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes
Paul Jung, Greg Markowsky
Ann. Probab. 41(4): 2682-2708 (July 2013). DOI: 10.1214/12-AOP770


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.

Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629–667] 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 Cohen and Samorodnitsky [Ann. Appl. Probab. 16 (2006) 1432–1461], we also consider alternating random reward schema associated to random walks at random times. Whereas random reward schema scale to local time fractional stable motions, we show that the alternating random reward schema 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 $H<1/2$.


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Paul Jung. Greg Markowsky. "Random walks at random times: Convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes." Ann. Probab. 41 (4) 2682 - 2708, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1306.60038
MathSciNet: MR3112928
Digital Object Identifier: 10.1214/12-AOP770

Primary: 60F05 , 60G22 , 60G52

Keywords: fractional Brownian motion , iterated process , local time fractional stable motion , random reward schema , Random walk in random scenery , self-similar process

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • July 2013
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