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. it appears that connecting the notion of long range dependence to certain types of phase transitions is promising. It fits well with intuition of the term “long memory” describing a model that is out of the ordinary. Furthermore, it allows us to concentrate on the behavior of really important functionals. Much remains to be done to clarify both possible types of such phase transitions and the relevant boundaries for concrete families of stochastic processes."[1]
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[2] and Mandelbrot and Wallis.[3] It is also due to the influence of these early papers and subsequent publications of Mandelbrot (especially Mandelbrot[4]) that long range dependence has also become associated with scaling and fractal behavior.
References
- ↑ GENNADY SAMORODNITSKY LONG RANGE DEPENDENCE Cornell (2007?) https://people.orie.cornell.edu/gennady/techreports/LRD-NOW.pdf
- ↑ B. Mandelbrot (1965): Une classe de processus stochastiques homothetiques a soi; application a loi climatologique de H.E. Hurst. Comptes Rendus Acad. Sci. Paris 240:3274–3277
- ↑ B. Mandelbrot and J. Wallis: Noah, Joseph and operational hydrology. Water Resources Research 4:909–918 (1986)
- ↑ B. Mandelbrot : The Fractal Geometry of Nature. W.H. Freeman and Co., San Francisco (1983)
