Let X1, X2,...be identically distributed random variables from an unknown continuous distribution. Further let Ir(1), Ir(2),...be a sequence of indicator functions defined on X1, X2,...by Ir(k) = 0 if k < r, Ir(k) = 1 if Xk is a r-record AND = 0 otherwise. Suppose that we observe X1, X2,... at times T1 < T2 <... where the Tk's are realisations of some regular counting process (N([tau])) defined on the positive half-line. Having observed [0, [tau]], say, the problem is to predict the future behaviour of the counting processes (Rr([tau], s))s[greater-or-equal, slanted][tau] = # r-records in [[tau], s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N([tau]))[tau][greater-or-equal, slanted]0 has dependent increments. The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.
