Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm（LIL for short）. For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.