# Statistics

Pink noise or 1⁄f noise (sometimes also called flicker noise) is a signal or process with a frequency spectrum such that the power spectral density (energy or power per Hz) is inversely proportional to the frequency.[jargon] In pink noise, each octave (halving/doubling in frequency) carries an equal amount of noise power. The name arises from the pink appearance of visible light with this power spectrum.[1] Within the scientific literature the term pink noise is sometimes used a little more loosely to refer to any noise with a power spectral density of the form where f is frequency and 0 < α < 2, with exponent α usually close to 1. These pink -like noises occur widely in nature and are a source of considerable interest in many fields.
Pink noise

There is a chapter on ``Coupling from the past,'' by James G. Propp and David B. Wilson, Chapter 22 of the textbook ''Markov Chains and Mixing Times,'' by David A. Levin, Yuval Peres, and Elizabeth L.
Web Site for Perfectly Random Sampling with Markov Chains:

The term "Lévy flight" was coined by Benoît Mandelbrot,[1] who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution,[2] and Rayleigh flight for when the distribution is a normal distribution[3] (which is not an example of a heavy-tailed probability distribution). Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.[4][5] A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which have a certain probability distribution, with the directions of the steps being isotropic and random. The particular case for which Mandelbrot used the term "Lévy flight"[1] is defined by the survivor function (commonly known as the survival function) of the distribution of step-sizes, U, being[6]
Lévy flight

Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics Conference. 3-5 June 1999. mathestate Web site for working on quantitative real estate problems, including some sections on stable distributions. For learning and theoretical experimentation purposes, choose Tools -> Tutorial Tools -> Tool #4 “Risk, Variation, and Tail Behavior”. For estimating stable parameters from your own data set, choose Tools -> Hands On Tools -> Tool #7 “Stable Data Analysis”. Mathematica package for stable distributions.
John Nolan's Stable Distribution Page

Autoregressive conditional heteroskedasticity
In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the error terms will have a characteristic size, or variance. In particular ARCH models assume the variance of the current error term or innovation to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations.

Nakagami distribution
The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter and a second parameter controlling spread, Characterization[edit]