Laws of large numbers for \(L\)-statistics
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 \(I\), such that the ``superposition operator'' defined by \(\Phi\) maps \(\mathbb{B}\) into \(\mathbb{L}_ 1(I, \text{Var }g)\). The result states that if \(\mathbb{F}_ n\) satisfies S(W)LLN in \(\mathbb{B}\), then \(L\) satisfies the same. Some theorems involving \(``\mathbb{L}_{p,q}\) spaces \(\mathbb{B}\)'', a corollary based on them, completed by another result, and a partial converse are also proved.
scientific article; zbMATH DE number 622170
scientific article