  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 $H<1/2$.
