Nonconstant levels for \(T\)-year return periods for two-dimensional Poisson processes with periodic intensity (Q1344828)

Assume that the time of events and their magnitudes \(\{(X_ n, Y_ n), n\geq 1\}\) form the points of a two-dimensional Poisson process with the times of events themselves forming a one-dimensional nonhomogeneous Poisson process with periodic intensity function. For each integer \(n\geq 1\), let \(M_ n= \sup\{Y_ i; X_ i\in I_ n\}\), where \(I_ n= [(n- 1)p, np)\) is the \(n\)th ``yearly'' interval. Each \(M_ n\) represents the maximum observed magnitude in the \(n\)th year. Define the thinned counting measure \(\xi(\cdot)= \sum \delta(X_ i, \cdot) I[Y_ i> u_ T(X_ i)]\) so that \(\xi(I_ n)= 0\) implies \(M_ n\leq u_ T(X_ i)\) for each \(X_ i\in I_ n\). The author presents a method for approximating a function \(u_ T(x)\) such that \(u_ T(x)= u_ T(x+ p)\) for each \(x\geq 0\) and \(P(\xi(I_ n)= 0)= 1- 1/T\) for each \(n\) and some fixed \(T> 1\). The function \(u_ T(x)\) is a nonconstant seasonal boundary which forms an upper bound on the process within each year with probability \(1- 1/T\).