General expressions are derived for the correlation coefficients between the length of an opening and that of the nth subsequent opening for a single ion channel. Analogous results are given for the correlation between shut times, and between an open time and subsequent shut times. An alternative derivation of the results of Fredkin et al. (in Proc. Berkeley Conf. in honor of Neyman & Kiefer, vol. 1, pp. 269-289 (1985] is given, and their results are extended to the case where openings occur in bursts. Expressions are given for the correlation between the first and nth opening in a burst, between the lengths of bursts, and between the number of openings per burst. Each of these sorts of correlation can give information about the connections that exist between the various states of the system; interpretations of the correlations are discussed. Expressions are derived for the distributions of the nth open time, shut time, burst length, etc. following the application of a perturbation (e.g. a voltage jump or a concentration jump). It is shown that these distributions will all be the same (namely the equilibrium distribution) only in the case where the openings, burst lengths, etc. are not correlated. Certain reaction schemes predict a component in the distribution of the number of openings per burst that has a unit mean (i.e. a component of isolated single openings). For some schemes this component is predicted to have zero amplitude, in principle, whereas in others it may be quite prominent. The presence or absence of this component can give information about the way in which the various states of the system are connected. The interpretation in terms of mechanism is discussed.