ABSTRACT
Although this book is concerned with discrete processes, the subject of the present chapter follows closely on the theory of stable continuous processes. The most commonly encountered model for continuous variables is based on the Gaussian probability density function. The reader is referred to the many texts on Gaussian variables and stochastic processes for details. Here we merely note that when independent Gaussian variables are added, then fluctuations in their sum are also characterised by a Gaussian probability density, a property known as statistical stability. In fact it has been known for more than 70 years that Gaussian is but one of a whole class of distributions governing the fluctuations of a variable with this property. In the case of continuous Gaussian noise, the property of statistical stability has greatly facilitated the mathematical description and analysis of a wide range of practical problems in all fields of science and engineering. However, other members of the class of stable distributions have recently begun to find applications on recognition of the importance of intermittency and other physical phenomena leading to outlying rare events. The relatively large probability of such events requires distributions with longer tails than Gaussian and other members of the stable class provide possible candidates for modelling data with such characteristics. These distributions have power-law tails +νp x x( ) ~ 1 1 with the index in the range 0 < ν < 2 so that the variance of the distributions is infinite. The special case ν = 2 gives the more familiar Gaussian case
∝ −p x ax( ) exp( )2 for which all moments exist. The property of long tails and the associated high probability of outly-
ing rare events can also be important attributes of discrete distributions. Although mention was made of such distributions in Chapter 2, Section 2.2, none of the processes described in earlier chapters are characterised by longtailed distributions. In this chapter, therefore, we shall develop a simple stochastic population model, based on multiple immigrations, for which the number probability distribution PN decreases at large values of N in inverse proportion to the number raised to a power. This will enable us to calculate the evolutionary behaviour of a simple Markov population exhibiting
extreme fluctuations in the number of individuals. Since the moments of the number distribution and the number correlation functions of the model do not exist, we shall investigate other measurable properties that will allow its fluctuation properties to be characterised.
