Laws of large numbers in self-correcting point processes (Q1086914)

The self-correcting point processes N(t), \(t\geq 0\), considered in this paper are assumed to be generated by conditional intensities \(E\{N(t+dt)- N(t)| {\mathcal F}_ t\}/dt\) of the form \(\psi\) (t-N(t)). (Here \({\mathcal F}_ t=\sigma \{N(s)\), \(0\leq s\leq t\})\). It is shown that if \(\psi\) satisfies certain conditions and X(t) denotes the Markov process t-N(t), then X(n), \(n=1,2,..\). is positive recurrent, and this is used to establish a law of large numbers for weighted averages of the form \(\int^{T}_{0}w(t,T)h(X(t))dt.\) These results extend those of \textit{D. Vere-Jones} and \textit{Y. Ogata} [J. Appl. Probab. 21, 335-342 (1984; Zbl 0548.60050)] where it was assumed that \(\psi\) (x) is an exponential \(\exp (a+bx)\).