On the order law of iterated logarithm.(Ukrainian, English)Zbl 1050.60006

Teor. Jmovirn. Mat. Stat. 68, 86-94 (2003); translation in Theory Probab. Math. Stat. 68, 93-102 (2004).
The author deals with the law of iterated logarithm for random variables from Banach lattices. One of the obtained results is the following: Let a separable Banach lattice $$B$$ does not contain $$l^{n}_{\infty}$$ and let $$X_{n}, n\geq1$$, be independent random elements in $$B$$ with zero mean and identical mean square deviation $$\Delta X$$, $$V_{n}=\sum_{i=1}^{n}b_{i}^2\to\infty$$, as $$n\to\infty$$. Let exist a positive element $$u\in B$$ and a sequence of positive numbers $$(M_{n})$$, such that $$M_{n}= o((V_{n}/ LL(V_{n}))^{1/2})$$, $$| b_{n} x_{n}|\leq M_{n}u$$ a.s., where $$L(t)=\max(1,\ln(t))$$. Then the order law of the iterated logarithm holds true: $\lim_{n\to\infty}\sup{\sum_{i=1}^{n}b_{i}X_{i}\over \chi(V_{n})}=\Delta X,\quad \lim_{n\to\infty}\inf{\sum_{i=1}^{n}b_{i}X_{i}\over \chi(V_{n})}=-\Delta X\quad \text{almost surely},$
where $$\chi(t)=(2tLL(t))^{1/2}$$.

MSC:

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