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The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. [ 1 ] [ 2 ] Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel).
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy , this distribution, with frequency as the dependent variable, is known as a van der Waals profile .
The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. [ 2 ] Of the three parameters defining the distribution, the stability parameter α {\displaystyle \alpha } is most important.
The counterpart of the stable distribution in this case is the geometric stable distribution Max-stability : here the operation is to take the maximum of a number of random variables. The counterpart of the stable distribution in this case is the generalized extreme value distribution , and the theory for this case is dealt with as extreme ...
The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It has parameter, α, which is defined over the range 0 < α ≤ 2, and where the case α = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions ...
The Laplace distribution; The Lévy skew alpha-stable distribution or stable distribution is a family of distributions often used to characterize financial data and critical behavior; the Cauchy distribution, Holtsmark distribution, Landau distribution, Lévy distribution and normal distribution are special cases. The Linnik distribution
Since -stable distributions have infinite -th moments for all >, the tempered stable processes have been proposed for overcoming this limitation of the stable distribution. On the other hand, GARCH models have been developed to explain the volatility clustering. In the GARCH model, the innovation (or residual) distributions are assumed to be a ...
The stretched exponential is also the characteristic function, basically the Fourier transform, of the Lévy symmetric alpha-stable distribution. In physics, the stretched exponential function is often used as a phenomenological description of relaxation in disordered systems.