Limit distributions for point processes of exceedances of random levels (Q1302068)

Assume that the sequence of r.v.'s \( X_n \) is stationary, the r.v. \( T \) is independent of \( \{ X_n \} \), and \( u_n(.)\) are measurable functions. The main result states sufficient conditions for the convergence in distribution of the point processes of exceedance \(S_n = \sum_{i=1}^n I_{ \{X_i > u_n(T)\} }\delta_{i/n}\). Then the limit point process is a double stochastic compound Poisson process. In the last section the author gives two interesting examples to illustrate the main statement and its corollary.