Complex probabilities on R^N as real probabilities on C^N and an application to path integrals

A necessary and sufficient condition for averages over complex-valued weight functions on RN to be represented as statistical averages over real, non-negative probability weights on CN was presented. It was shown that many path integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates. For particular case of N = 1, it was shown that the condition was fulfilled for all complex weights normalized to a total integral of 1.