The time series of a gamma-ray burst (GRB) is a non-stationary time series and all of its statistical properties vary with time. Considering that each GRB is a different manifestation of the same stochastic process, we have studied the time-dependent and time-averaged probability density functions (pdfs), which characterize the underlying stochastic process. The pdfs are fitted with a Gaussian distribution function. It has been argued that the Gaussian pdfs possibly indicate the turbulent origin of GRBs. The spectral and temporal evolutions of GRBs are also studied using the evolution of spectral forms, colour–colour diagrams and hysteresis loops. The results do not contradict the interpretation of the turbulence of GRBs.