On the law of the iterated logarithm in Banach lattices

The law of the iterated logarithm in the classical form
$$ \limsup_{n\to\infty}\frac{X_1+X_2+\cdots+X_n}{(2n\log\log(n))^{1/2}}=\mathfrak{G} X $$
is established for some Banach lattices.