  We consider continuous time random walks (CTRW) for open systems that
exchange energy and matter with multiple reservoirs. Each waiting time
distribution (WTD) for times between steps is characterized by a positive
parameter a, which is set to a=1 if it decays at least as fast as t^{-2} at
long times and therefore has a finite first moment. A WTD with a<1 decays as
t^{-a-1}. A fluctuation theorem for the trajectory quantity R, defined as the
logarithm of the ratio of the probability of a trajectory and the probability
of the time reversed trajectory, holds for any CTRW. However, R can be
identified as a trajectory entropy change only if the WTDs have a=1 and satisfy
separability (also called "direction time independence"). For nonseparable WTDs
with a=1, R can only be identified as a trajectory entropy change at long
times, and a fluctuation theorem for the entropy change then only holds at long
times. For WTDs with 0<a<1 no meaningful fluctuation theorem can be derived. We
also show that the (experimentally accessible) n'th moments of the energy and
matter transfers between the system and a given reservoir grow as t^{n a} at
long times.
