Self-similar Processes Research Articles

Hierarchies of Higgs Field and Theoretical Calculation of Higgs Particle Mass

A concept of isotime spin for Higgs particle and fermions is proposed in this paper. The Higgs field may be the time field, and the electroweak phase transition corresponds to the time phase transition. By describing the self-similar transformation process of the time phase transition, it is proved that the structural details of the time phase transition fully meet the requirements of the Higgs mechanism. Theoretical formulas of the Higgs mass, the mass and isotime spins of fermions are derived. The exact critical temperature of the electroweak phase transition is calculated, by using the solution for critical point and the experimental value of the Higgs particle mass. We discuss the behavior of the free Higgs particle and the saturation state of the time field containing I-bosons, point out that there is no parallel universes because weak force destroys the self-similar transformation of the time field, and the probability of the time field energy as a type of dark energy.

Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised kth order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised kth order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

MONITORING OF LINK-LEVEL CONGESTION IN TELECOMMUNICATION SYSTEMS USING INFORMATION CRITERIA

The successful functioning of telecommunication networks largely depends on the effectiveness of algorithms for detection and protection against overloads. The article describes the main differences that arise when forecasting, monitoring and managing congestion at the node level and at the channel level. An algorithm for detecting congestion by estimating the entropy of time distributions of traffic parameters is proposed. The entropy measures of data sets for various types of model distribution, in particular for the Pareto distribution, which optimally describes the behavior of self-similar random processes, were calculated and analyzed. The advantages of this approach include scalability, sensitivity to changes in distributions of traffic characteristics and ease of implementation and accessible interpretation.

Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions

In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although the two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly α-stable process at its first exit time from (0,∞). We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.

On the local times of noise reinforced Bessel processes

We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in(0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

Discrete self-similar and ergodic Markov chains

The first aim of this paper is to introduce a class of Markov chains on Z+ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway intertwining relation between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on R+. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains which are nonreversible. By means of intertwining relations and their strengthened interweaving versions, we derive several deep analytical properties of such ergodic chains, including the description of the spectrum, the spectral expansion of their semigroups and the study of their convergence to equilibrium in the Φ-entropy sense as well as their hypercontractivity property.

Multi-Fractality, Universality and Singularity in Turbulence

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all turbulent flows. Finer observations however prove that this picture is too simplistic, owing to intermittency of energy dissipation and high velocity derivatives. In this review, we discuss how such intermittency can be explained and fitted into a new picture of turbulence. We first discuss how the concept of multi-fractality (invented by Parisi and Frisch in 1982) enables to generalize the concept of self-similarity in a non-homogeneous environment and recover a universality in turbulence. We further review the local extension of this theory, and show how it enables to probe the most irregular locations of the velocity field, in the sense foreseen by Lars Onsager in 1949. Finally, we discuss how the multi-fractal theory connects to possible singularities, in the real or in the complex plane, as first investigated by Frisch and Morf in 1981.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions.

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

We consider a class of generalized time-fractional evolution equations containing a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

Entropic Compressibility of Lévy Processes

In contrast to their seemingly simple and shared structure of independence and stationarity, L\'evy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari (2018), we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a L\'evy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on L\'evy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of L\'evy processes, captured by their Blumenthal-Getoor index.

Self-similarity and response of fractional differential equations under white noise input

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann–Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t→at (a>0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coefficient is a2H being H>0 the Hurst index. In the practical applications, the phenomena are usually ruled by fractional differential equations involving more terms. In this paper it is shown that the response of a multi-term fractional differential equation is a linear combination of self-similar processes with increasing order of Hurst exponent. The consequences of self-similarity are discussed in detail, closed forms of correlation and variance are presented for the general case and particularized for the cases of engineering interest.

SCATTERING STATISTICS OF GENERALIZED SPATIAL POISSON POINT PROCESSES.

We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.

Generalized Wiener–Hermite integrals and rough non-Gaussian Ornstein–Uhlenbeck process

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval . We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process.

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

We study the fluctuations of the area under a self-similar Gaussian process x(τ) with Hurst exponent H > 0 (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate r. Typical fluctuations of A(t) scale as for large t and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of A(t). In the long-time limit t → ∞, we find that the full distribution of the area takes the form with anomalous exponents α = 1/(2H + 2) and β = (2H + 3)/(4H + 4) in the regime of moderately large fluctuations, and a different anomalous scaling form in the regime of very large fluctuations. The associated rate functions Φ(y) and Ψ(w) depend on H and are found exactly. Remarkably, Φ(y) has a singularity that we interpret as a first-order dynamical condensation transition, while Ψ(w) exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of Φ(y) around the origin y = 0 correctly describes the typical, Gaussian fluctuations of A(t). Despite these anomalous scalings, we find that all of the cumulants of the distribution grow linearly in time, , in the long-time limit. For the case of reset Brownian motion (corresponding to H = 1/2), we develop a recursive scheme to calculate the coefficients c n exactly and use it to calculate the first six nonvanishing cumulants.

Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation

We study the asymptotics of the k-regular self-similar fragmentation process. For α>0 and an integer k≥2, this is the Markov process (It)t≥0 in which each It is a union of open subsets of [0,1), and independently each subinterval of It of size u breaks into k equally sized pieces at rate uα. Let k−mt and k−Mt be the respective sizes of the largest and smallest fragments in It. By relating (It)t≥0 to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that |mt−g(t)|≤1 and |Mt−h(t)|≤1 for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size k−n exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as n→∞.

Scaling laws for properties of random graphs that grow via successive combination

Abstract We consider undirected graphs that grow through the successive combination of component sub-graphs. For any well-behaved functions defined for such graphs, taking values in a Banach space, we show that there must exist a scaling law applicable when successive copies of the same component graph are combined. Crucially, we extend the approach introduced in previous work to the successive combination of component random sub-graphs. We illustrate this by generalizing the preferential attachment operation for the combination of stochastic block models. We discuss a further wide range of random graph combination operators to which this theory now applies, indicating the ubiquity of growth scaling laws (and asymptotic decay scaling laws) within applications, where the modules are quite distinct, yet may be considered as instances drawn from the same random graph. This is a type of statistically self-similar growth process, as opposed to a deterministic growth process incorporating exact copies of the same motif, and it represents a natural, partially random, growth processes for graphs observed in the analysis of social and technology contexts.

Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters

We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is introduced, based on which we obtain approximately asymptotically consistent algorithms for clustering locally asymptotically self-similar stochastic processes. In the simulation study, clustering data sampled from fractional and multifractional Brownian motions with distinct Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets’ equity indexes returns and sovereign CDS spreads provides a successful real world application. Implementations in MATLAB of the proposed algorithms and the simulation study are publicly shared in GitHub.

ANALYSIS OF MONOLITHIC AND MICROSERVICE ARCHITECTURES FEATURES AND METRICS

In this paper the information technologies stack is presented. Thesetechnologies are used during network architecture deployment. The analysis of technological advantages and drawbacks under investigation for monolithic and network architectures will be useful during of cyber security analysis in telecom networks. The analysis of the main numeric characteristics was carried out with the aid of Kubectl. The results of a series of numerical experiments on the evaluation of the response speed to requests and the fault tolerance are presented. The characteristics of the of monolithic and microservice-based architectures scalability are under investigation. For the time series sets, which characterize the network server load, the value of the Hurst exponent was calculated.
 The research main goal is the monolithic and microservice architecture main characteristics analysis, time series data from the network server accruing, and their statistical analysis.
 The methodology of Kubernetes clusters deploying using Minikube, Kubectl, Docker has been used. Application deploy on AWS ECS virtual machine with monolithic architecture and on the Kubernetes cluster (AWS EKS) were conducted.
 The investigation results gives us the confirmation, that the microservices architecture would be more fault tolerance and flexible in comparison with the monolithic architecture. Time series fractal analysis on the server equipment load showed the presence of long-term dependency, so that we can treat the traffic implementation as a self-similar process.
 The scientific novelty of the article lies in the application of fractal analysis to real time series: use of the kernel in user space, kernel latency, RAM usage, caching of RAM collected over 6 months with a step of 10 seconds, establishing a long-term dependence of time series data.
 The practical significance of the research is methodology creation of the monolithic and microservice architectures deployment and exploitation, as well as the use of time series fractal analysis for the network equipment load exploration.

Growth-fragmentation process embedded in a planar Brownian excursion

The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane $${\mathbb {H}}$$ , obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in (Bertoin et al. Probab Theory Relat Field 172:663–724, 2018).