On the Law of the Iterated Logarithm in Banach Lattices

One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

“…In [2], this estimate was used in the proof of the order law of the iterated logarithm and somewhat strengthened for B = R 1 , namely, it was shown that…”

“…Recall the corresponding definition (for details, see [2]). A continuous function ψ(t) defined on R 1 , even, convex, and positive for t = 0 is called an N -function [9, p. 149] if lim t→0 t −1 ψ(t) = 0 and lim t→∞ t −1 ψ(t) = +∞.…”

“…Equalities (3) are called the order law of the iterated logarithm in a Banach lattice B. The case b i = 1 was considered in [2].…”

“…Inequalities (1),(2), and (4) resemble the classical Khinchin inequality. However, in contrast to the latter, the opposite inequality in (1),(2), and (4) does not hold in the general case for any constant C. Indeed, in inequality (1) for the real axis, we setx i = χ(2 2n ) for i = 2 2nand x i = 0 for i = 2 2n .…”

Ordinal law of the iterated logarithm in Banach lattices and some applications

“…The main aim of this paper is to study a new variant of the law of the iterated logarithm, called the ordinal law of the iterated logarithm, for random elements assuming values in a Banach lattice. The definition of the ordinal law of the iterated logarithm is introduced in [7]. We need some notions of the theory of Banach lattices (see [8]- [10]) to recall the definition of the ordinal law of the iterated logarithm.…”