Maxima with random indexes
Let \(\{\xi_i\}\) and \(\{X_i\}\) be two sequences of r.v.'s, where \(\{X_i\}\) is i.i.d. Put \(M_n=\bigvee_{i=1}^n \xi_i\) and \(\widehat M_n=\bigvee_{i=1}^n X_i.\) Let \(U\) be a regularly varying function (at \(+\infty\)) with index \(\rho\) (write \(U\in \text{RV}_\rho\)). Let \(\eta\) be a positive r.v. and \(U\in\text{RV}_\rho\) \((\rho>0).\) Suppose that for any real numbers \(u_n\) \[ \big| P\big( M_n\leq u_n\mid \eta=y\big)-P\big( \widehat M_{[U(n)]}\leq u_n\mid\eta=y\big) \big|\longrightarrow 0,\qquad n\to\infty, \] for \(y>0.\) Let \(\{N_n\}\) be a positive integer-valued r.v. such that \(N_n/n {\buildrel P\over\rightarrow}\eta\). If there exist \(a_n>0,\) \(b_n\in{\mathbb R},\) \(n\geq 1,\) such that \[ P\big( M_n\leq a_nx+b_n \big) {\buildrel W\over\longrightarrow} G(x),\quad n\uparrow\infty, \] where \(G\) is nondegenerate distribution function, then \[ P\big( M_{N_n}\leq a_{[U(n)]}x+b_{[U(n)]} \big) {\buildrel W\over\longrightarrow} \int_0^{+\infty}\big( G(x) \big)^{y^\rho}dP(\eta\leq y),\quad n\uparrow\infty. \]
scientific article; zbMATH DE number 1159090