Law of the iterated logarithm for m-dependent random variables under sub-linear expectations

Determine whether analogues of Theorem 2 and Chen’s limit-form law of the iterated logarithm hold for m-dependent random variables under sub-linear expectations.

Background

The paper establishes a limit-form law of the iterated logarithm for independent and identically distributed random variables in Peng’s sub-linear expectation framework. In particular, Theorem 2 shows that the limiting standard-deviation parameter can be a random function of the observations, and the associated compact law of the iterated logarithm can have a symmetric random interval as its set of limit points.

The authors note that related strong laws of large numbers and laws of the iterated logarithm have recently been proved for m-dependent random variables under sub-linear expectations. They explicitly leave unresolved whether the stronger conclusions obtained in the paper, or the earlier limit-form law of the iterated logarithm, extend from the independent setting to m-dependent sequences.