  A time trajectory of an observable that fluctuates between two values (say,
on and off), stemming from some unknown multi-substate kinetic scheme, is the
output of many single molecule experiments. Here we show that when all
successive waiting times along the trajectory are uncorrelated the on and the
off waiting time probability density functions (PDFs) contain all the
information. By relating the lack of correlation in the trajectory to the
topology of kinetic schemes, we can immediately specify those kinetic schemes
that are equally consistent with experiment, which means that it is impossible
to differentiate between them by any sophisticated analyses of the trajectory.
Correlated trajectories, however, contain additional information about the
underlying kinetic scheme, and we consider the strategy that one should use to
extract it. An example is given on correlations in the activity of individual
lipase molecules.
