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Theoretical calculations, simulations and measurements of rotation of earthquake focal mechanisms suggest that the stress in earthquake focal zones follows the Cauchy distribution which is one of the stable probability distributions (with the value of the exponent α equal to 1). We review the properties of the stable distributions and show that the Cauchy distribution is expected to approximate the stress caused by earthquakes occurring over geologically long intervals of a fault zone development. However, the stress caused by recent earthquakes recorded in instrumental catalogues, should follow symmetric stable distributions with the value of α significantly less than one. This is explained by a fractal distribution of earthquake hypocentres: the dimension of a hypocentre set, δ, is close to zero for short-term earthquake catalogues and asymptotically approaches 2¼ for long-time intervals. We use the Harvard catalogue of seismic moment tensor solutions to investigate the distribution of incremental static stress caused by earthquakes. The stress measured in the focal zone of each event is approximated by stable distributions. In agreement with theoretical considerations, the exponent value of the distribution approaches zero as the time span of an earthquake catalogue (ΔT) decreases. For large stress values α increases. We surmise that it is caused by the δ increase for small inter-earthquake distances due to location errors.