Laws of large numbers for \(L\)-statistics
type: Item
instance of: scholarly article
title: Laws of large numbers for \(L\)-statistics
zbMATH Open document ID: 0823.60028
DOI: 10.1155/S1048953394000146
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1995-10-08T00:00:00Z
full work available at URL: https://eudml.org/doc/47269
review text: The problem considered is simple to describe. Namely, let \(U_ 1,\dots, U_ n\) be \(\lambda\) (i.e. uniformly) distributed on (0,1), \(U_{n,1} \leq \dots \leq U_{n,n}\) be their order statistics, let \(\int | J| d\lambda < \infty\), \(c_{ni}(J) = \int \chi_{[(i - 1)/n, i/n]} Jd\lambda\) and \(L_ n(J,g) = n^{-1} \sum^ n_{i = 1} c_{ni} (J) g(U_{ni})\), with \(g : (0,1) \to \mathbb{R}\). Find conditions under which \(L_ n\) converges to \(\int Jgd \lambda\) a.s. (SLLN) or in probability (WLLN). The first result gives sufficient conditions: \((g (U_{n,1}) c_{n1} + g(U_{n,n})c_{nn})/n \to 0\) a.s. and in probability, respectively, together with \(\int g_ M | J| d\lambda < \infty\), for \(M = 0\) in the first case and for all \(M \in (0,1)\) in the second one. For \(M > 0\), \(g_ M(t)\) is the supremum of \(| g|\) over the interval \([Mt, t/M]\) for \(t < 1/2\) and \([1-(1 - t)/M, 1 - M(1 - t)]\) for \(t \geq 1/2\), for \(M = 0\) it has another analogous expression involving log etc. However, the methods of obtaining the results, which we consider as remarkable, are too difficult to describe in detail in a review; many of the 22 quotations are effectively used. We get an idea by mentioning a more general result, in which the \(U\)'s have a general distribution \(F\), supposed to have a median \(c\) not charged by \(F\); consider the smallest interval \(I = (a,b)\) with \(F(\mathbb{C} \overline{I}) = 0\). Consider a function \(\Phi_ J(t,x)\); its definition is \(\int \chi_{[| m(t)|, \min(| x|, \gamma)]} (s) J_ c(s,t)ds\), for a fixed \(\gamma \in (1/2, 1)\), where \(J_ c (s,t) = J(s) \chi_{(a,c)} (t) + J(1 -s)\chi_{[c,b)} (t)\) and \(m(t) = E \mathbb{S}_ n (t)\), \(\mathbb{S}_ n(t) = -\mathbb{F}_ n (t)\) and \(1 - \mathbb{F}_ n(t)\) for \(t \in (a,c)\), \(t \in [c,b)\), \(\mathbb{F}_ n(t)\) being the empirical distribution function. Consider a ``quasi-Banach'' function space \(\mathbb{R}\) on \(I\) with respect to the measure \(\text{Var }g\), supposed to be Radon on \(I\)…
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 622170
zbMATH Keywords: empirical distribution
zbMATH Keywords: superposition operator
zbMATH Keywords: quasi Banach function space
zbMATH Keywords: symmetrization
author: Rimas Norvaiša
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
OpenAlex ID: W2149135443
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description: scientific article; zbMATH DE number 622170