On the properties of random multiplicative measures with the multipliers   exponentially distributed
  Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized dimensions, and negative values of generalized dimensions arise. Three equivalent methods of classification of the random multifractal measures are proposed, which is based on: (i) the discrepancy between the curves of generalized dimensions, (ii) the solution properties of equation T(qcrit) =0, and (iii) the relative position of the curve f(alpha) and the diagonal f(alpha)=alpha in the first quadrant. These three classes correspond to \mu([0,1])=infinity, \mu([0,1])=1 and \mu([0,1])=0, respectively. Phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated. 
Wei-Xing Zhou and Zun-Hong yu