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The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables. Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution.
Given a linear time-invariant (LTI) system represented by a nonsingular matrix , the relative gain array (RGA) is defined as = = (). where is the elementwise Hadamard product of the two matrices, and the transpose operator (no conjugate) is necessary even for complex .
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).
The probability density function is (,) = ((+)) (),where I 0 (z) is the modified Bessel function of the first kind with order zero.. In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter =, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter = +, defined as the total power ...
The multivariate stable distribution defines linear relations between stable distribution marginals. [clarification needed] In the same way as for the univariate case, the distribution is defined in terms of its characteristic function. The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution.
That is, if x belongs to the interior of its stable manifold, it is asymptotically stable if it is both attractive and stable. (There are examples showing that attractivity does not imply asymptotic stability. [9] [10] [11] Such examples are easy to create using homoclinic connections.)
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1] [2]In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is
In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting. An equilibrium solution f e {\displaystyle f_{e}} to an autonomous system of first order ordinary differential equations is called: