We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

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In the modern world of "Big Data," dynamic signals are often multivariate and characterized by joint scale-free dynamics (self-similarity) and non-Gaussianity. In this paper, we examine the performance of joint wavelet eigenanalysis estimation for the Hurst parameters (scaling exponents) of non-Gaussian multivariate processes. We propose a new process called operator fractional Lévy motion (ofLm) as a Lévy-type model for non-Gaussian multivariate self-similarity. Based on large size Monte Carlo simulations of bivariate ofLm with a combination of Gaussian and non-Gaussian marginals, the estimation performance for Hurst parameters is shown to be satisfactory over finite samples.

The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a canonical model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in wide a range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology to the analysis of geophysical flow data shows that tfBm provides a much closer fit than fBm.where u + = u1 {u>0} and B(du) is an independently scattered Gaussian random measure satisfying EB(du) 2 = σ 2 du. In the boundary case λ = 0 (and 0 < H < 1), tfBm reduces to a fractional Brownian motion (fBm), namely, a Gaussian, stationary-increment, self-similar process (e.g., Embrechts and Maejima (2002), Taqqu (2003)). TfBm is a new canonical model introduced in Meerschaert and Sabzikar that displays the so-named Davenport spectrum (Davenport (1961)). The latter is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for low frequency behavior and has * The second author was partially supported by the prime award no. W911NF-14-1-0475 from the Biomathematics subdivision of the Army Research Office, USA. The authors would like to thank Laurent Chevillard for his comments on this work. †