The authors describe a method of analysis of random signals containing non-Gaussian features such as pulses or oscillatory bursts; the method involves detecting a pulse locating it in time and classifying it in height and then averaging the signal over a fixed time interval centred on the pulse and over many occurrences of pulses in each class. They show that the technique retains information which is lost in the usual two-point correlation function measurements: this information is that contained in the non-random phase relationships between different Fourier modes, which are necessarily entailed by the non-Gaussian distribution. They also show that in suitable cases the technique allows the direct detection of some kinds of non-linear effect. They illustrate the method with examples from fluctuation measurements in magnetically confined plasmas.