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In this paper, we study a class of self-exciting point processes. Theintensity of the point process has a nonlinear dependence on the past historyand time. When a new jump occurs, the intensity increases and we expect morejumps to come. Otherwise, the intensity decays. The model is a marriage betweenstochasticity and dynamical system. In the short-term, stochasticity plays amajor role and in the long-term, dynamical system governs the limiting behaviorof the system. We study the law of large numbers, central limit theorem, largedeviations and asymptotics for the tail probabilities.

Asymptotics for a Class of Self-Exciting Point Processes