The article shows that the concept of pre-envelopes can be applied to random processes that are not stationary. When a real random process and its Hilbert transform are considered, they have the same autocovariance function and zero crosscovariance at the same time. The autocovariance function of the pre-envelope is twice that of the original process. Time-dependent power density spectrum helps understand non-stationary bandpass processes. The autocovariance function of non-stationary bandpass processes can be derived from that of stationary ones with simple adjustments.