https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

S. Bochner's concept of a subordinate stochastic process is proposed as a model for speculative price series. A general class of finite-variance distributions for price changes is described, and a member of this class, the lognormal-normal, is tested against previously proposed distributions for speculative price differences. It is shown with both discrete Bayes' tests and Kolmogorov-Smirnov tests that finite-variance distributions subordinate to the normal fit cotton futures price data better than members of the stable family.

/pdf/a-subordinated-stochastic-process-model-with-finite-variance-4jw5uaoy6u.pdf

A subordinated stochastic process model with finite variance for speculative prices

Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc.

We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.

/pdf/a-riemannian-framework-for-tensor-computing-mitwgnphxc.pdf

A Riemannian Framework for Tensor Computing

We examine the detrended fluctuation analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling which appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.

Detecting long-range correlations with detrended fluctuation analysis

Ein physikalischer Prozes (die Anderung eines physikalischen Systems) heist stochastisch-definit, wenn aus der Kenntnis des ZustandesX0 des Systems in einem gewissen Zeitmomentt0 die Kenntnis der Verteilungsfunktion der Wahrscheinlichkeiten fur die moglichen ZustandeX des Systems in einem Zeitmomentt>t0 folgt.

J. Marshall

Papers

Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)