Rescaled range analysis is a statistical methodology used to analyse long records of natural phenomena and to detect the presence or absence of trend in time series by finding Hurst component. Hurst exponent is an important measurement in fractal analysis, descriptions of rough images and signals. Basically this method is used to identify when the event is persistent i.e., the tendency to continue its current direction and also anti persistent i.e, the tendency to reverse itself or it is random and unpredictable. In this study, Hurst__ampersandsignlsquo;s rescaled range technique was applied to a time series composed of seismic events occurred in Gujarat for a period of 1973 to 2010.In this paper, we studied the earthquake cycles and the results infer that these have the behaviour of the so called ?Joseph effect__ampersandsignlsquo;__ampersandsignlsquo;: quiet years tend to be followed by quiet years and active years by active years. The Hurst exponent was found to be 0.469, which is nearly 0.5