Search results
Results from the WOW.Com Content Network
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.
The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
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).
Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.
Von Neumann stability analysis is a commonly used procedure for the stability analysis of finite difference schemes as applied to linear partial differential equations. These results do not hold for nonlinear PDEs, where a general, consistent definition of stability is complicated by many properties absent in linear equations.
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.
Steady-states can be stable or unstable. A steady-state is unstable if a small perturbation in one or more of the concentrations results in the system diverging from its state. In contrast, if a steady-state is stable, any perturbation will relax back to the original steady state. Further details can be found on the page Stability theory.
More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.