Random walks at random times: convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes

Given a random walk (Sn)n∈Z defined for a doubly infinite sequence of times, we let the time parameter (nk)k∈N itself be a process with values in Z and call (Snk)k∈N a random walk at ran-dom time. We show that under suitable conditions, it scales to an (H-sssi)-time Lévy motion, a generalization of iterated Brownian motion. In Khoshnevisan and Lewis (1999), a normalized variation result for iterated Brownian motion was stated which alternated between even and odd orders of variation. This result “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 (2006), we also consider alternating random reward schema (random reward schema are certain sums of random walks in random scenery) associated to random walks at random times. Random reward schema scale to local time fractional stable motions which are H-sssi symmetric α-stable processes with self-similarity exponents H> 1/α. We show that the alternating random reward schema scale to indicator fractional stable motions which are H-sssi symmetric α-stable processes with self-similarity exponents H < 1/α. We also show that one may use a recursion on the subordinating 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 α = 2, the fractional stable motions given by the recursion are fractional Brownian motions with dyadic H ∈ (0, 1). Finally, we show that “undoing a subordination ” via a time-change allows one to, in some sense, extract Brownian motion from fractional Brownian motions with H < 1/2