An upper bound on the binomial process approximation to the exceedance process (Q1424683)

The authors deal with the approximation to the point process of exceedances over a higher threshold. Given \(n\) i.i.d. random variables with common distribution function \(F\), the exceedances over a threshold \(u<\omega(F)\) can be distributionally described by a binomial point process \(N_0:=\sum_{i\leq K(n)}{\mathcal E}_{X_{i}}\), where \(\omega(F)\) is the upper endpoint of the support of \(F\); \(X_{1},X_{2},\ldots\) are i.i.d. random variables with common distribution function \(F^{[u]}(x)=[F(x)-F(u)]/\bar F(u)\), \(x\geq u\), \(\overline F(u)=1-F(u)\); \(K(n)\) is a binomial random variable with parameters \(n\) and \(p:=\overline F(u)\) which is independent of the sequence \(X_{1},X_{2},\ldots\). The original binomial process \(N_0\) is replaced by a binomial process \(N_1\), where the original random variables \(X_{i}\) are replaced by i.i.d. random variables \(Y_{i}\) with a common generalized Pareto distribution function \(W_{\gamma,u,\sigma(u)}\) with the shape, location and scale parameters \(\gamma,u,\sigma(u)\). The upper bound on the remainder term of the approximation is established in terms of the von Mises function \(\eta(t):= ({1-F\over f})'(F^{-1}(1-t))-\gamma\) for some real parameter \(\gamma\). The authors establish upper bounds on the ultimate and penultimate approximations which are determined by the von Mises function \(\eta.\)