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

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 $$E| \xi|^{1/p}<\infty$$.

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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