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

Matsak, I. K.

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 \(\chi(t)=(2tLL(t))^{1/2}\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60B12

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

Keywords:

order law of iterated logarithm; Banach space; random elements