Aleksandre Shangua

Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

In terms of almost surely convergence in weak topology in a Banach space a characterization of finite-dimensionality of the space is obtained. An analogy of Kolomogorov's three series theorem for the weak topology convergence in Banach spaces is given

The following results are announced: If a sequence of random elements tea normed space converges to zero in probability, then there exists a permutation of the given sequence, which is Cesáro convergent almost surely. Every bounded in L, sequence of random variables allows a permutation under which it satisfies the strong law of large numbers. A statement about Banach spaces with the Banach-Saks property is also given.

The permutational and the weak permutational Banach-Saks properties are introduced and their relationship with the usual Banach-Saks and Weak Banach-Saks properties is clarified