Stable process

Abstract: THE large-scale dynamical properties of some physical systems depend on the dynamical evolution of a large number of nonlinearly coupled subsystems. Examples include systems that exhibit self-organized criticality1 and turbulence2,3. Such systems tend to exhibit spatial and temporal scaling behaviour–power–law behaviour of a particular observable. Scaling is found in a wide range of systems, from geophysical4 to biological5. Here we explore the possibility that scaling phenomena occur in economic systemsa-especially when the economic system is one subject to precise rules, as is the case in financial markets6–8. Specifically, we show that the scaling of the probability distribution of a particular economic index–the Standard & Poor's 500–can be described by a non-gaussian process with dynamics that, for the central part of the distribution, correspond to that predicted for a Levy stable process9–11. Scaling behaviour is observed for time intervals spanning three orders of magnitude, from 1,000 min to 1 min, the latter being close to the minimum time necessary to perform a trading transaction in a financial market. In the tails of the distribution the fall-off deviates from that for a Levy stable process and is approximately exponential, ensuring that (as one would expect for a price difference distribution) the variance of the distribution is finite. The scaling exponent is remarkably constant over the six-year period (1984-89) of our data. This dynamical behaviour of the economic index should provide a framework within which to develop economic models.

Abstract: Examples of stable laws in applications Analytic properties of the distributions in the family $\mathfrak S$ Special properties of laws in the class $\mathfrak W$ Estimators of the parameters of stable distributions.

Abstract: The Stable Distribution Symmetric Stable Random Processes Covariation and Conditional Expectation Parameter Estimates for Symmetric Stable Distributions Estimation of Covariations Parametric Models of Stable Processes Linear Theory of Stable Processes Symmetric Stable Models for Impulsive Noise Signal Detection in Stable Noise Current and Future Trends in Signal Processing with Alpha-Stable Distributions.

Abstract: The problem of tracking curves in dense visual clutter is a challenging one. Trackers based on Kalman filters are of limited use; because they are based on Gaussian densities which are unimodal, they cannot represent simultaneous alternative hypotheses. Extensions to the Kalman filter to handle multiple data associations work satisfactorily in the simple case of point targets, but do not extend naturally to continuous curves. A new, stochastic algorithm is proposed here, the Condensation algorithm — Conditional Density Propagation over time. It uses ‘factored sampling’, a method previously applied to interpretation of static images, in which the distribution of possible interpretations is represented by a randomly generated set of representatives. The Condensation algorithm combines factored sampling with learned dynamical models to propagate an entire probability distribution for object position and shape, over time. The result is highly robust tracking of agile motion in clutter, markedly superior to what has previously been attainable from Kalman filtering. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time.