Stochastic Process with Memory

Full Title

The notion of long range dependence has, clearly, something to do with memory in a stochastic process.. The “specialness” of long memory indicates that most stationary stochastic processes do not have it.

A stationary process is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.

Clearly a photon that as originated in the early universe has a very long memory.

Context

The first serious argument that this can be important is in a series of papers of B. Mandelbrot and his co-authors, e.g. Mandelbrot (1965) and Mandelbrot and Wallis (1968). It is also due to the influence of these early papers and subsequent publications of Mandelbrot (especially Mandelbrot (1983)) that long range dependence has also become associated with scaling and fractal behavior.