  Peng (2006) initiated a new kind of central limit theorem under sub-linear
expectations. Song (2017) gave an estimate of the rate of convergence of Peng's
central limit theorem. Based on these results, we establish a new kind of
almost sure central limit theorem under sub-linear expectations in this paper,
which is a quasi sure convergence version of Peng's central limit theorem.
Moreover, this result is a natural extension of the classical almost sure
central limit theorem to the case where the probability is no longer additive.
Meanwhile, we prove a new kind of strong law of large numbers for non-additive
probabilities without the independent identically distributed assumption.
