Boundary crossing random variables related to quantile convergence (Q1096960)

Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with distribution F. Suppose that \(0<p<1\), that \(\xi_ p\) is the unique p-th quantile of F, and that \(\xi_{p,n}\) is the sample p-th quantile of \(X_ 1,...,X_ n\). If \(b(n)\to 0^+\) sufficiently slowly then \[ N(b(n))=\sum^{\infty}_{n=1}I\{| \xi_{p,n}-\xi_ p| >b(n)\}\quad and \] \[ L(b(n))=\sup \{n:| \xi_{p,n}-\xi_ p| >b(n)\} \] are ty space and (\({\mathcal F}_ n)\) an increasing family of sub-\(\sigma\)-algebras of \({\mathcal F}\) with supremum \({\mathcal F}_{\infty}\). If \(\mu\) : \({\mathcal F}\to {\mathbb{R}}\) is a signed measure, \(\mu\) induces an (\({\mathcal F}_ n)_{n\in {\mathbb{N}}\cup \{\infty \}}\)-adapted process \((X_ n)\) in the following way. Let \(\mu_ n\) denote the restriction of \(\mu\) to \({\mathcal F}_ n\) and define \(X_ n\) to be the Radon-Nikodym derivative of the absolutely continuous part of \(\mu_ n\) relative to \(\lambda\). The Andersen-Jessen theorem states that lim \(X_ n=X_{\infty}^ a.\)s.. This result has been extended to vector-valued stochastic processes induced by finitely additive set functions or set function sequences. In the present paper the author studies in a quite general context structure and limit behaviour of such induced processes. He shows how the Andersen-Jessen theorem and its extensions can be located within the convergence theory for martingales and their generalizations.