scholarly journals Integral-geometric construction of self-similar stable processes

1991 ◽  
Vol 123 ◽  
pp. 1-12 ◽  
Author(s):  
Shigeo Takenaka
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Natural Science ◽  
Stable Processes ◽  
Self Similarity ◽  
Fractional Brownian Motions ◽  
Stationary Increments ◽  
Construction Of Self ◽  
Self Similar ◽  
Complex Phenomena

Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t є R} with stationary increments which has a self-similar property, that is, there exists a constant H (for the Brownian motion H = 1/2, in general 0 < H < 1 for Gaussian processes) called the exponent of self-similarity of the process, such that, for any c > 0, two processes are subject to the same law (see [10]).

scholarly journals The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Francesco Mainardi ◽  
Antonio Mura ◽  
Gianni Pagnini
Keyword(s):  
Diffusion Processes ◽  
Integral Operators ◽  
Fractional Diffusion ◽  
Wright Function ◽  
Fractional Brownian Motions ◽  
Stationary Increments ◽  
Time Integral ◽  
Gaussian Density ◽  
Self Similar ◽  
Derivatives Of

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as -Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the -Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.

scholarly journals Double hypergeometric Lévy processes and self-similarity

2021 ◽  
Vol 58 (1) ◽  
pp. 254-273
Author(s):  
Andreas E. Kyprianou ◽  
Juan Carlos Pardo ◽  
Matija Vidmar
Keyword(s):  
Closed Form ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stable Processes ◽  
Self Similarity ◽  
New Family ◽  
Self Similar

AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

AN INTRODUCTION TO THE THEORY OF SELF-SIMILAR STOCHASTIC PROCESSES

2000 ◽  
Vol 14 (12n13) ◽  
pp. 1399-1420 ◽  
Author(s):  
PAUL EMBRECHTS ◽  
MAKOTO MAEJIMA
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
High Energy Physics ◽  
High Energy ◽  
Point Of View ◽  
Long Range Dependence ◽  
Self Similar ◽  
Energy Physics

Self-similar processes such as fractional Brownian motion are stochastic processes that are invariant in distribution under suitable scaling of time and space. These processes can typically be used to model random phenomena with long-range dependence. Naturally, these processes are closely related to the notion of renormalization in statistical and high energy physics. They are also increasingly important in many other fields of application, as there are economics and finance. This paper starts with some basic aspects on self-similar processes and discusses several topics from the point of view of probability theory.

ANALYSIS OF THE PROCESS OF SELFSIMILARITY OF NETWORK TRAFFIC AS AN APPROACH TO DETECTING CYBER ATTACKS ON COMPUTER NETWORKS

2020 ◽  
Author(s):  
И.В. КОТЕНКО ◽  
А.М. КРИБЕЛЬ ◽  
О.С. ЛАУТА ◽  
И.Б. САЕНКО
Keyword(s):  
Brownian Motion ◽  
Computer Networks ◽  
Network Traffic ◽  
Telecommunication Network ◽  
Cyber Attacks ◽  
Self Similarity ◽  
Data Set ◽  
Fractal Brownian Motion ◽  
Self Similar

Предложен подход кобнаружению кибератак на компьютерные сети, основанный на выявлениианомалий в сетевом трафике путем оценки свойства самоподобия. Рассмотрены методы выявления долговременной зависимости в фрактальном броуновском движении и реальном сетевом трафике компьютерных сетей. Показано, что трафик телекоммуникационной сети является самоподобной структурой и его поведение близко к фрактальному броуновскому движению. В качестве инструментов при разработке данного подхода были использованы фрактальный анализ и математическая статистика. Анализируются вопросы программной реализации предлагаемого подхода и формирования набора данных, содержащего сетевые пакеты компьютерных сетей. Экспериментальные результаты, полученные с использованием сгенерированного набораданных, продемонстрировали наличие самоподобия у сетевого трафика компьютерных сетей и подтвердили высокую эффективность предлагаемого подхода: он позволяет обнаруживать кибератаки в реальном или близком к реальному масштабе времени. The paper discusses an approach to detecting cyber attacks on computer networks, based on identifying anomalies in network traffic by assessing its self-similarity property. Methods for identifying long-term dependence in fractal Brownian motion and real network traffic of computer networks are considered. It is shown that the traffic of a telecommunication network is a self-similar structure and its behavior is close to fractal Brownian motion. Fractal analysis and mathematical statistics were used as tools in the development of this approach. The issues of the software implementation of the proposed approach and the formation of a data set containing network packets of computer networks are considered. The experimental results obtained using the generated dataset demonstrated the existence of selfsimilarity in the network traffic of computer networks and confirmed the fair efficiency of the proposed approach. The proposed can be used to quickly detect cyber attacks in real or near real time.

scholarly journals (1α)-Self similar α-stable processes with stationary increments

1990 ◽  
Vol 35 (2) ◽  
pp. 308-313 ◽  
Author(s):  
Gennady Samorodnitsky ◽  
Murad S. Taqqu
Keyword(s):  
Stable Processes ◽  
Stationary Increments ◽  
Self Similar
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