Random record processes and state dependent thinning

Suppose that a point process N̄t = T1, T2, … if [0, ∞) is thinned by independently retaining Tn with probability pn. Our main examples are the classical p-thinning (pn ≡ p) and the random record process (pn = 1n). When N̄t is a mixed, nonhomogeneous Poisson process, we find conditions under which the thinned process is Poisson. When N̄t is a pure birth process (gamma-mixed Poisson with exponential rate), we show that the record process is Markov renewal, with an interesting structure, and we compare this with related asymptotic results. When N̄t is a Mittag-Leffler renewal process (the homogeneous Poisson is a special case), we give a “Deheuvels-type” representation of the record process (Deheuvels, 1982) and related characterization results.