Self-similar Stochastic Processes Research Articles

A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics

In this paper, we present a general mathematical construction that allows us to define a parametric class of H-sssi stochastic processes (self-similar with stationary increments), which have marginal probability density function that evolves in time according to a partial integro-differential equation of fractional type. This construction is based on the theory of finite measures on functional spaces. Since the variance evolves in time as a power function, these H-sssi processes naturally provide models for slow- and fast-anomalous diffusion. Such a class includes, as particular cases, fractional Brownian motion, grey Brownian motion and Brownian motion.

Fractional Lévy motion through path integrals

Fractional Lévy motion (fLm) is the natural generalization of fractional Brownian motion in the context of self-similar stochastic processes and stable probability distributions. In this paper we give an explicit derivation of the propagator of fLm by using path integral methods. The propagators of Brownian motion and fractional Brownian motion are recovered as particular cases. The fractional diffusion equation corresponding to fLm is also obtained.

Extracting features of Gaussian self-similar stochastic processes via the Bandt-Pompe approach

By recourse to appropriate information theory quantifiers (normalized Shannon entropy and Martín-Plastino-Rosso intensive statistical complexity measure), we revisit the characterization of Gaussian self-similar stochastic processes from a Bandt-Pompe viewpoint. We show that the ensuing approach exhibits considerable advantages with respect to other treatments. In particular, clear quantifiers gaps are found in the transition between the continuous processes and their associated noises.

Self-similar stochastic processes in solar wind turbulence

Solar wind data is used to estimate the autocorrelation function for the stochastic process x( τ) = y( t + τ) − y( t), considered as a function of τ, where y( t) is any one of the quantities B 2( t), n p( t) V 2( t), or n p( t). This process has stationary increments and a variance that increases like a power law τ 2 γ where γ is the scaling exponent. For the kinetic energy density and the proton density the scaling exponent is close to the Kolmogorov value γ = 1/3, for the magnetic energy density it is slightly larger. In all three cases, it is shown that the autocorrelation function estimated from the data agrees with the theoretical autocorrelation function for a self-similar stochastic process with stationary increments and finite variance. This is far from proof, but it suggests that these stochastic processes may be self-similar for time scales in the small scale inertial range of the turbulence, that is, from approximately 10 to 10 3 s.

Parameter estimation of selfsimilarity exponents

The characteristic feature of operator selfsimilar stochastic processes is that a linear rescaling in time is equal in the sense of distributions to a linear operator rescaling in space, which in turn is characterized by the selfsimilarity exponent. The growth behaviour of such processes in any radial direction is determined by the real parts of the eigenvalues of the selfsimilarity exponent. We extend an estimation method of Meerschaert and Scheffler [M.M. Meerschaert, H.-P. Scheffler, Moment estimator for random vectors with heavy tails, J. Multivariate Anal. 71 (1999) 145–159, M.M. Meerschaert, H.-P. Scheffler, Portfolio modeling with heavy tailed random vectors, in: S.T. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance, Elsevier Science B.V., Amsterdam, 2003, pp. 595–640] to be applicable for estimating the real parts of the eigenvalues of the selfsimilarity exponent and corresponding spectral directions given by the eigenvectors. More generally, the results are applied to operator semi-selfsimilar processes, which obey a weaker scaling property, and to certain Ornstein–Uhlenbeck type processes connected to operator semi-selfsimilar processes via Lamperti's transformation.

Characterization of Gaussian self-similar stochastic processes using wavelet-based informational tools

Efficient tools to characterize stochastic processes are discussed. Quantifiers originally proposed within the framework of information theory, like entropy and statistical complexity, are translated into wavelet language, which renders the above quantifiers into tools that exhibit the important "localization" advantages provided by wavelet theory. Two important and popular stochastic processes, fractional Brownian motion and fractional Gaussian noise, are studied using these wavelet-based informational tools. Exact analytical expressions are obtained for the wavelet probability distribution. Finally, numerical simulations are used to validate our analytical results.

On use of the alpha stable self-similar stochastic process to model aggregated VBR video traffic

The alpha stable self-similar stochastic process has been proved an effective model for high variable data traffic. A deep insight into some special issues and considerations on use of the process to model aggregated VBR video traffic is made. Different methods to estimate stability parameter α and self-similar parameter H are compared. Processes to generate the linear fractional stable noise (LFSN) and the alpha stable random variables are provided. Model construction and the quantitative comparisons with fractional Brown motion (FBM) and real traffic are also examined. Open problems and future directions are also given with thoughtful discussions.

Operator self-similar processes on Banach spaces

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form {sH:s>0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH:s>0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Mellin Convolution for Subordinated Stable Processes

In this paper, we shall show how to interpret (3) by means of the convolution integral of the Mellin transform, and how to use the tools of the Mellin-Barnes integrals to treat the subordination for the class of self-similar stochastic processes, which are governed by Levy strictly stable probability distributions. The paper is divided as follows. In Sec. 2, we provide the reader with the essential notions and notations concerning the Mellin transforms. In particular, we recall the fundamental property of the Mellin convolution to represent the p.d.f. of the product of two independent random variables and consequently to interpret the subordination formula involving a self-similar parent process. In Sec. 3, we first recall the Mellin-Barnes integral representation of the whole class of Levy (symmetric and not symmetric) strictly stable densities, as derived in previous papers (3, 5). Then, through the Mellin convolution, we derive for these densities a subordination formula involving, as directing p.d.f.'s, those of unilateral stable densities. In so doing, we clarify and generalize known results of Feller, proven via Fourier and Laplace transforms. Finally, in Sec. 4, we draw the main conclusions and outline the direction for future work.

Phase-modulated resonances modeled as self-similar processes with application to turbulent sounds

In this paper, we propose a nonlinear stochastic model for time-varying resonances where the instantaneous phase (and frequency) of a sinusoidal oscillation is allowed to vary proportionally to an /spl alpha/-stable self-similar stochastic processes. The main motivation of our work stems from previous experimental and theoretical evidence that speech resonances in fricative sounds can be modeled phenomenologically as AM-FM signals with randomly varying instantaneous frequencies and that several signal classes related to turbulent phenomena are self-similar 1/f processes. Our general approach is to model the instantaneous phase of an AM-FM resonance as a self-similar /spl alpha/-stable process. As a special case, this random phase model includes the class of random fractal signals known as fractional Brownian motion. We theoretically explore this random modulation model and analytically derive its autocorrelation and power spectrum. We also propose an algorithm to fit this model to arbitrary resonances with random phase modulation. Further, we apply the above ideas to real speech data and demonstrate that this model is suitable for resonances of fricative sounds.

Probabilistic envelope processes for α-stable self-similar traffic models and their application to resource provisioning

Recent experimental studies have shown that α-stable self-similar stochastic processes can accurately characterize various types of aggregate network traffic. Using this traffic modeling approach, we propose some probabilistic envelope processes that can be used to represent the resource demand of a traffic stream in performance evaluation studies. We illustrate the use of the proposed envelopes in resource allocation for data and video traffic and in the design of an admission control mechanism. From our analysis we conclude that the presence of heavy tails in the distribution of a traffic process has a severe impact on the dimensioning of network elements.

Stochastic properties of the linear multifractional stable motion

We study a family of locally self-similar stochastic processes Y = {Y(t)} t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

Stochastic properties of the linear multifractional stable motion

We study a family of locally self-similar stochastic processes Y = {Y(t)}t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

Elliptic Self-Similar Stochastic Processes

Elliptic Self-Similar Stochastic Processes

Local self-similarity and the Hausdorff dimension

Let X be a locally self-similar stochastic process of index 0< H<1 whose sample paths are a.s. C H− ε for all ε>0. Then the Hausdorff dimension of the graph of X is a.s. 2− H. To cite this article: A. Benassi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Discrete random walk models for space–time fractional diffusion

Abstract A physical–mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space–time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2] and skewness θ (|θ|⩽min{α,2−α}), and the first-order time derivative with a Caputo derivative of order β∈(0,1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

A new heavy-tailed discrete distribution for LRD M/G/∞ sample generation

Several traffic measurement reports have convincingly shown the presence of self-similarity at the packet level in current networks, inducing as a result a revolution in the stochastic modeling of traffic. The essence of this behavior can be adequately captured by several classes of self-similar stochastic processes. But the use of these in performance analysis has opened new problems and research issues, also in simulation studies where the efficient generation of synthetic sample paths with self-similar properties is one of the fundamental concerns. In this paper, we present a flexible and efficient generator of self-similar traces, based on a simple M/G/∞ model which uses a new heavy-tailed discrete distribution.

A Computation Method for Providing Statistical Performance Guarantee to an ATM Switch 1

ATM is a connection-oriented technology and messages are divided into fixed size packet called cells to facilitate communications. However, before two hosts can communicate, a connection has to be established between them. Consider a real-time communication application running on top of an ATM network. In order to provide the real-time service, we require the network to provide a performance guarantee for the connection. There are two types of performance guarantees: deterministic and statistical guarantees. While a deterministic guarantee provides an absolute bound on the worst case cell delay experienced in an ATM switch, a statistical guarantee provides a probabilistic bound on the worst case cell delay. In this paper, we use a self-similar stochastic process to characterize the arrival of the real-time traffic. Extending from our previous work on deterministic delay guarantee, we provide methods for determinating the statistical delay bound for the worst case cell delay in an ATM switch with various output port controllers. We conclude this paper with two case studies: one based on ordinary LAN traffic and one on the variable bit-rate MPEG video transmission of the movie “Star Wars”. To show the effectiveness of our statistical delay guarantee, we compare it with the delay derived by Cruz as well as the actual cell delay determined by the two traffic traces.

Use of α-stable self-similar stochastic processes for modeling traffic in broadband networks

In this article, we propose a new model for aggregate network traffic. This model, besides reflecting self-similarity and long-range dependence, is able to capture the appropriate level of burstiness of different types of traffic by selecting the proper parameters. Different types of self-similar traffic traces (LAN/WAN, WWW, VBR video) are analysed by estimating their self-similarity coefficient H, as well as the parameters of their marginal distributions. When comparing the real traces with our artificial traces, the agreement, which was evaluated both qualitatively (visually) and quantitatively (by means of the marginal CDF and the periodogram), is better than that achieved with previously proposed models. By analysing different types of traffic traces, the model is shown to be flexible enough to be applied to a variety of communications scenarios. A queue with our proposed traffic as input is analysed. A proof of convergence of aggregate traffic to α-stable processes is also included, as well as the conditions under which the Gaussian assumption is appropriate.

Self-Similarity of Mathematical Likelihood

We study sufficiency in terms of the extent to which the Taylor series expansion of the normed log-likelihood function generated by the sampling distribution of a statistic conforms to that of the sample normed log-likelihood function, and give an associated self-similar property of mathematical likelihood (not self-similarity in the sense of self-similar stochastic processes).