On the law of the iterated logarithm in Banach lattices (Q2711158)

A Banach lattice \(E\) is called \(q\)-concave, \(1\leq q <\infty,\) if there exists a constant \(D_{(q)}=D_{(q)}(E)\) such that for any \(n\) and any elements \((x_i)^n_1\subset E\) NEWLINE\[NEWLINE \Biggl(\sum_{i=1}^n\|x_i\|^q \Biggr)^{1/q}\leq D_{(q)} \left\|\Biggl(\sum_{i=1}^n|x_i|^q \Biggr)^{1/q}\right\|.NEWLINE\]NEWLINE For \(q\)-concave Banach lattices satisfying some additional conditions the author proves the law of the iterated logarithm in the classical form NEWLINE\[NEWLINE \lim\sup_{n\to\infty}{ X_1+\cdots+X_n\over (2n\log\log n)^{1/2}}={\mathcal G} X, NEWLINE\]NEWLINE where \({\mathcal G} X\) is the mean square deviation of the random element \(X\).