Random walks at random times: Convergence to iterated L\'{e}vy motion, fractional stable motions, and other self-similar processes

TL;DR

This paper investigates the scaling limits of random walks at random times, revealing connections to iterated Lévy motions, fractional stable motions, and self-similar processes, and introduces recursive mechanisms to generate new stable processes.

Contribution

It generalizes the concept of iterated Brownian motion to broader stable processes and explores recursive constructions of self-similar stable motions at random times.

Findings

Random walks at random times converge to iterated Lévy motions.

Alternating random reward schema scale to indicator fractional stable motions.

Recursive subordination produces new stable processes and links to Brownian motion.

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. 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 $H<1/2$.