Hyperproperties have shown to be a powerful tool for expressing and reasoning about information-flow security policies. In this paper, we investigate the problem of statistical model checking (SMC) for hyperproperties. Unlike exhaustive model checking, SMC works based on drawing samples from the system at hand and evaluate the specification with statistical confidence. The main benefit of applying SMC over exhaustive techniques is its efficiency and scalability. To reason about probabilistic hyperproperties, we first propose the temporal logic HyperPCLT* that extends PCTL* and HyperPCTL. We show that HyperPCLT* can express important probabilistic information-flow security policies that cannot be expressed with HyperPCTL. Then, we introduce SMC algorithms for verifying HyperPCLT* formulas on discrete-time Markov chains, based on sequential probability ratio tests (SPRT) with a new notion of multi-dimensional indifference region. Our SMC algorithms can handle both non-nested and nested probability operators for any desired significance level. To show the effectiveness of our technique, we evaluate our SMC algorithms on four case studies focused on information security: timing side-channel vulnerability in encryption, probabilistic anonymity in dining cryptographers, probabilistic noninterference of parallel programs, and the performance of a randomized cache replacement policy that acts as a countermeasure against cache flush attacks.