  In this paper, a framework on a discrete observation of (marked) point
processes under the high-frequency observation is developed. Based on this
framework, we first clarify the relation between random coefficient
integer-valued autoregressive process with infinite order (RCINAR($\infty$))
and i.i.d.-marked self-exciting process, known as marked Hawkes process. For
this purpose, we show that the point process constructed of the sum of a
RCINAR($\infty$) converge weakly to a marked Hawkes process. This limit theorem
establish that RCINAR($\infty$) processes can be seen as a discretely observed
marked Hawkes processes when the observation frequency increases and thus build
a bridge between discrete-time series analysis and the analysis of
continuous-time stochastic process and give a new perspective in the point
process approach in extreme value theory. Second, we give a necessary and
sufficient condition of the stationarity of RCINAR($\infty$) process and give
its random coefficient autoregressive (RCAR) representation. Finally, as an
application of our results, we establish a rigorous theoretical justification
of self-exciting peaks over threshold (SEPOT) model, which is a well-known as a
(marked) Hawkes process model for the empirical analysis of extremal events in
financial econometrics and of which, however, the theoretical validity have
rarely discussed. Simulation results of the asymptotic properties of
RCINAR($\infty$) shows some interesting implications for statistical
applications.
