The Mandelbrot Set with Partial Memory

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA

Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical dat...

A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951-1980

Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical dat...

Conditional distributions of Mandelbrot-Van Ness fractional Lévy processes and continuous-time ARMA-GARCH-type models with long memory

Long memory effects can be found in different kind of data from finance to hydrology. Therefore, models which can reflect these properties have become more popular in recent years especially in the fields of time series analysis, econometrics and financial mathematics. Mandelbrot-Van Ness fractional Lévy processes allow for such stationary long memory effects in their increments and have been u...

Quadratic polynomials, multipliers and equidistribution

Given a sequence of complex numbers ρn, we study the asymptotic distribution of the sets of parameters c ∈ C such that the quadratic maps z2+c has a cycle of period n and multiplier ρn. Assume 1 n log |ρn| → L. If L ≤ log 2, they equidistribute on the boundary of the Mandelbrot set. If L > log 2 they equidistribute on the equipotential of the Mandelbrot set of level 2L− 2 log 2. Introduction In...