  In this paper, we establish some general forms of the law of the iterated
logarithm for independent random variables in a sub-linear expectation space,
where the random variables are not necessarily identically distributed.
Exponential inequalities for the maximum sum of independent random variables
and Kolmogorov's converse exponential inequalities are established as tools for
showing the law of the iterated logarithm. As an application, the sufficient
and necessary conditions of the law of iterated logarithm for independent and
identically distributed random variables under the sub-linear expectation are
obtained.

Probability