Order-type law of iterated logarithm in Banach lattices and some its applications. (Ukrainian, English) Zbl 1150.60002

Matsak, I. K.

Teor. Jmovirn. Mat. Stat. 74, 68-80 (2006); translation in Theory Probab. Math. Stat. 74, 77-91 (2007).

The author obtains necessary and sufficient conditions for fulfilment of the order-type law of iterated logarithm in Banach lattices. Let \(X\) be a random element with values in separable Banach lattice \(B\). Let us define \(G_{p}X=\sup(X\in K_{p}(X))\), where \(K_{p}(X)=\{E(\eta X):\;E|\eta|^{q}\leq 1\}\), \(1/p+1/q=1\), \(G(X)=G_2(X)\), \(E\) - is the Pettice integral. One of the presented results is the following.
Let \(B\) be a separable \(q\)-concave Banach lattice, \(1\leq q<2\), let \(X\) be a random element with values from \(B\), \(EX=0\). The random element \(X\) satisfies order-type law of iterated logarithm if and only if there exist \(GX\) in \(B\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60B12

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

Keywords:

order-type law of iterated logarithm; Banach lattices; empiric processes