A Glivenko-Cantelli theorem and strong laws for \(L\)-statistics
Consider the Marcinkiewicz-Zygmund strong law (MZSLLN) which states the almost sure convergence of \(n^{-1}\sum^n_{i=1} X_i=c+o (n^{(1-p)/p})\) a.s. as \(n\to\infty\), for \(\{X_i, i\geq 1\}\) i.i.d. random variables with \(E| X_1|^p<\infty\) for some \(0<p<2\). This paper establishes the law MZSLLN for \(L\)-statistics. It takes a similar approach to Wellner's SLLN and LIL for \(L\)-statistics and first establishes a suitable Glivenko-Cantelli type theorem (Theorem 1). This is then used to obtain the MZSLLN for \(L\)-statistics (Theorem 2). Let \({\mathfrak G}\) be the set of left continuous functions on (0,1) which are of bounded variation on \((\theta,1-\theta)\) for any \(\theta>0\). Consider some constants \(c_{n1},\dots,c_{nn}\) \((n\geq 1)\) and sequences of functions \(g_n\in {\mathfrak G}\). The sequence of \(L\)-statistics is defined by \(L_n=n^{-1}\sum^n_{i=1} c_{ni}g_n (\xi_{ni})\), where \((\xi_i,\;i\geq 1)\) are i.i.d. uniformly distributed random variables on (0,1). Define \(J_n(t)=c_{ni}\) if \((i-1)/n<t\leq i/n\), \(1\leq i \leq n\), and \(J_n(0)=c_{n1}\). Let \(\mu_n=\int^1_0g_nJ_ndt\). For real numbers \(b_1, b_2, \delta\), define \(B_{b_1,b_2}(t)= t^{-b_1} (1-t)^{-b_2}\), \(0<t<1\). Consider the following three conditions on \((g_n,\;n\geq 1)\) and \((J_n, n\geq 1)\). (1) \(| J_n|\leq B_{\gamma,\varepsilon}\) for some \(\gamma\) and \(\varepsilon\). (2) For some \((r,s)\) with \(-1<r+\gamma<1/p\) and \(-1<s+ \varepsilon<1/p\), \(| g_n|\leq ct^{-r} (1-t)^{-s}\). (3) \(\sup_{n\geq 1}\int B_{\gamma, \varepsilon}B_{\delta,\delta}hd| g_n|<\infty\), for some \(\delta >0\) and some \(h\) satisfying \(h(t)=h(1-t)\), for \(0\leq t\leq 1/2\) or nonnegative nondecreasing continuous function on [0,1]. The main theorem states: if \((g_n, J_n,n\geq 1)\) satisfies the conditions (1) -- (3) above, then \(L_n-\nu_n =o(n^{(1-p)/p})\) a.s. Several examples demonstrate Theorem 1 and Theorem 2.
scientific article; zbMATH DE number 1234274