Power law distributions have been repeatedly observed in a wide variety of
socioeconomic, biological and technological areas. In many of the observations,
e.g., city populations and sizes of living organisms, the objects of interest
evolve due to the replication of their many independent components, e.g.,
births-deaths of individuals and replications of cells. Furthermore, the rates
of the replication are often controlled by exogenous parameters causing periods
of expansion and contraction, e.g., baby booms and busts, economic booms and
recessions, etc. In addition, the sizes of these objects often have reflective
lower boundaries, e.g., cities do not fall bellow a certain size, low income
individuals are subsidized by the government, companies are protected by
bankruptcy laws, etc.
Hence, it is natural to propose reflected modulated branching processes as
generic models for many of the preceding observations. Indeed, our main results
show that the proposed mathematical models result in power law distributions
under quite general polynomial Gartner-Ellis conditions, the generality of
which could explain the ubiquitous nature of power law distributions. In
addition, on a logarithmic scale, we establish an asymptotic equivalence
between the reflected branching processes and the corresponding multiplicative
ones. The latter, as recognized by Goldie (1991), is known to be dual to
queueing/additive processes. We emphasize this duality further in the
generality of stationary and ergodic processes.Comment: 36 pages, 2 figures; added references; a new theorem in Subsection
4.