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For a random walk defined for a doubly infinite sequence of times, we let thetime parameter itself be an integer-valued process, and call the orginalprocess a random walk at random time. We find the scaling limit whichgeneralizes the so-called iterated Brownian motion. Khoshnevisan and Lewis[Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form ofmeasure-theoretic duality" between iterated Brownian motion and a Brownianmotion in random scenery. We show that a random walk at random time can beconsidered a random walk in "alternating" scenery, thus hinting at a mechanismbehind this duality. Following Cohen and Samorodnitsky [Ann. Appl. Probab. 16(2006) 1432-1461], we also consider alternating random reward schema associatedto random walks at random times. Whereas random reward schema scale to localtime fractional stable motions, we show that the alternating random rewardschema scale to indicator fractional stable motions. Finally, we show that onemay recursively "subordinate" random time processes to get new local time andindicator fractional stable motions and new stable processes in random sceneryor at random times. When $\alpha=2$, the fractional stable motions given by therecursion are fractional Brownian motions with dyadic $H\in(0,1)$. Also, we seethat "un-subordinating" via a time-change allows one to, in some sense, extractBrownian motion from fractional Brownian motions with $H<1/2$.

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