Ordered convergence and Glivenko-Kantelli type theorems in \(L_{p}(-\infty,\infty)\). (Ukrainian, English) Zbl 1164.60309

Matsak, I. K.

Teor. Jmovirn. Mat. Stat. 75, 71-79 (2006); translation in Theory Probab. Math. Stat. 75, 83-92 (2007).

Let \(\xi,\xi_1,\xi_2,\dots\in \mathbb R^1\) be a sequence of i.i.d. random variables with distribution function \(F(t)\). Let us define the empirical distribution function as \(F_{n}(t)={1\over n}\sum_{i=1}^{n}I_{(-\infty,t)}(\xi_{i})\). The sequence \((x_{n})\) of elements of a Banach lattice \(B\) is called an ordered converged to the element \(x\): \(x=o\text{-}\lim_{n\to\infty}x_{n}\), if there exists a sequence \((v_{n})\) such that \(| x-x_{n}|<v_{n}\), and \(v_{n}\downarrow0\). We say, that random elements \(X_{n}, n=1,2,\dots\), \(E[X_{n}]=0, \forall n\), satisfy the ordered law of large numbers if \(o\text{-}\lim_{n\to\infty}{1\over n}\sum_{i=1}^{n}X_{i}=0\) a.s. The author proves that, for \(1<p<\infty\), the empirical distribution function \(F_{n}(t)\) satisfies the ordered law of large numbers \(\lim_{m\to\infty}\int_{-\infty}^{\infty}|\sup_{n>m}F_{n}(t)-F(t)|^{p}\,dt=0\), \(\lim_{m\to\infty}\int_{-\infty}^{\infty}|\inf_{n>m}F_{n}(t)-F(t)|^{p}\,dt=0\) a.s. if and only if \(E| \xi|^{1/p}<\infty\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60B12

Limit theorems for vector-valued random variables (infinite-dimensional case)

Keywords:

ordered convergence; ordered law of large numbers; Glivenko-Kantelli theorems in \(L_{p}(-\infty,\infty)\)