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In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a…

We study the anomalous scaling of the mass density measure of Lagrangian tracers in a compressible flow realized on the free surface on top of a three dimensional flow. The full two dimensional probability distribution of local stretching…

Mathematical formulations and proofs for a wavelet based statistic employed in functional data analysis is elaborately discussed in this report. The propositions and derivations discussed here apply to a wavelet based statistic with hard…

We study a scenario under which variable step random walks give anomalous statistics. We begin by analyzing the Martingale Central Limit Theorem to find a sufficient condition for the limit distribution to be non-Gaussian. We note that the…

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and…

For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the…

We present a theoretical framework for understanding the wavefunctions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong…

To investigate the structure of individual differences in performance on behavioral tasks, Haaf and Rouder (2017) developed a class of hierarchical Bayesian mixed models with varying levels of constraint on the individual effects. The…

Random walks are studied on disordered cellular networks in 2-and 3-dimensional spaces with arbitrary curvature. The coefficients of the evolution equation are calculated in term of the structural properties of the cellular system. The…

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable L\'evy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time…

We propose a wavelet-based approach to construct consistent estimators of the pointwise H\"older exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our…

Experimental and empirical data are often analyzed on log-log plots in order to find some scaling argument for the observed/examined phenomenon at hands, in particular for rank-size rule research, but also in critical phenomena in…

We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression…

Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally…

We take a wavelet based approach to the analysis of point processes and the estimation of the first order intensity under a continuous time setting. A multiresolution analysis of a point process is formulated which motivates the definition…

We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal…

Random walks represent an important tool for probing the structural and dynamical properties of networks and modeling transport and diffusion processes on networks. However, when individuals' movement becomes dictated by more complicated…

The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due…

A direct numerical simulation of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.28 is carried out at moderate Reynolds number, simulating flow conditions similar to those of the experiment by Dupont et al.…

We propose a wavelet based method for the characterization of the scaling behavior of non-stationary time series. It makes use of the built-in ability of the wavelets for capturing the trends in a data set, in variable window sizes.…