Random walks at random times and their reward schema: weak convergence to iterated Levy motion, fractional stable motions, and other self-similar processes

Details Random walks at random times and their reward schema: weak convergence to iterated Levy motion, fractional stable motions, and other self-similar processes Random walks at random times and their reward schema: weak convergence to iterated Levy motion, fractional stable motions, and other self-similar processes

2011-05-16

arxiv.org/abs/1105.3130v3

Mathematics

Probability

24 pages, 1 figure. A new theorem has been added, and several typos have been corrected

Scientific paper

Given a random walk defined for doubly infinite times, let the time parameter n_k be a process with integer values and call S_{n_k} a random walk at random time. This process scales to an (H-sssi)-time alpha-stable Levy motion, a generalization of iterated Brownian motion. A result of Khoshnevisan and Lewis (1999) "suggested a measure-theoretic duality" between iterated Brownian motion and Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus giving a mechanism for this duality. Following Cohen and Samorodnitsky (2006), we also consider alternating random reward schema associated to random walks at random times. While random reward schema scale to local time fractional stable motions with H>1/alpha, the alternating random reward schema scale to indicator fractional stable motions with H<1/alpha. We also use a certain recursion to get new local time and indicator fractional stable motions. When alpha=2, the fractional stable motions given by the recursion are fractional Brownian motions with dyadic H. Finally when alpha=2, we show that a time-change allows one to extract Brownian motion from fractional Brownian motions with H<1/2.

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