On a Geometric Mean and Power-law Statistical Distributions

For a large class of statistical systems a geometric mean value of the ob-servables is constrained. These observables are characterized by a power-law statistical distribution. In everyday life we find events of very different nature often to follow similar statistical distributions. These distributions arise because the same stochastic process is at work, and this process can be understood beyond the context of each example. Who is responsible for the particular forms of these distributions? This is a fundamental problem. The least bias approach to this problem was promoted by E.T.Jaynes about half century ago as a Maximum Entropy Principle (MEP) [1]. The MEP states that the physical observable has a distribution, consistent with given constraints which maximize the entropy. It is important that the a priori given constraints are defined by a macroscopic mechanism the observable is constructed. If the observable values are restricted to be within a limited interval only, the resulting statistical distribution is a uniform distribution. A maximization of Shannon-Gibbs entropy with a constraint on the observable's arithmetic mean value results in an exponential distribution. Examples of the observ-ables constructed such a way that an arithmetic mean values are constrained are following.