Strong approximation of maxima by extremal processes
Let \(X_1,X_2,\dots\) be a sequence of i.i.d. random variables, \(M_n= \max_{1\leq k\leq n}X_k\) and \((M_n- b_n)/a_n@> D>>G\) for some \(a_n> 0\), \(b\in\mathbb{R}\), and a nondegenerate distribution function \(G\). Consider the sequence of \(D(0,\infty)\) stochastic processes \(\{Y_n\}\) given by \(Y_n(t)= (M_{[nt]}- b_{[nt]})/a_{[nt]}\) for \(t\geq 1/n\) and \(Y_n(t)= (X_1- b_1)/a_1\) for \(t< 1/n\). It is proved that weak invariance principle holds, i.e. \(Y_n@>D>> Y_{0,G}\), where the limiting process \(Y_{0,G}\) is a transformed extremal process and moreover \(Y_{0,G}(e^t)\) is a stationary Markov process. Next, several strong approximations of \(Y_n\) by \(Y_{0,G}\) are obtained for all types of \(G\) and with remainder tending to zero in probability or almost surely. The results extend those obtained by Deheuvels in a 1981--1983 series of papers.
scientific article; zbMATH DE number 1933036
scientific article