Bounded and compact laws of the logarithm for B-valued random variables

In this paper, we study a version of the law of the logarithm in a Banach space setting. Some necessary and some sufficient conditions are presented for the law of the logarithm for B-valued random variables. The law of the logarithm, the law of the iterated logarithm and the central limit theorem are shown to be equivalent for finite-dimentional B-valued random variables. However, this statement is not true for infinite-dimensional case. Under the central limit theorem, the law of the logarithm is shown to be equivalent to some certain moment condition. The law of the iterated logarithm implies the law of the logarithm, but the converse is not true. All methods used in this paper are quite standard in probability in Banach spaces except for some modifications. We made an effort to solve this problem completely in a Banach space using both the isoperimetric methods and the Gaussian randomization technique, but we were not successful.