Fractals are objects that possess a form of self-scaling; a part of the whole can be
made to recreate the whole by shifting and stretching. Many objects are self-scaling
only in a statistical sense, meaning that a part of the whole can be made to recreate the
whole in the likeness of their probability distributions, rather than as exact replicas.
Examples of random fractals include the length of a segment of coastline, the variation
of water flow in the river Nile, and the human heart rate.
Point processes are mathematical representations of random phenomena whose
individual events are largely identical and occur principally at discrete times and
locations. Examples include the arrival of cars at a tollbooth, the release of neuro-
transmitter molecules at a biological synapse, and the sequence of human heartbeats.
Fractals began to find their way into the scientific literature some 50 years ago.
For point processes this took place perhaps 100 years ago, although both concepts
developed far earlier. These two fields of study have grown side-by-side, reflecting
their increasing importance in the natural and technological worlds. However, the
domains in which point processes and fractals both play a role have received scant
attention. It is the intersection of these two fields that forms the topic of this treatise.
Fractal-based point processes exhibit both the scaling properties of fractals and
the discrete character of random point processes. These constructs are useful for
representing a wide variety of diverse phenomena in the physical and biological
sciences, from information-packet arrivals on a computer network to action-potential
occurrences in a neural preparation.