scholarly journals Multifractal Analysis of Infinite Products of Stationary Jump Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-26
Author(s):  
Petteri Mannersalo ◽  
Ilkka Norros ◽  
Rudolf H. Riedi
Keyword(s):  
Stochastic Processes ◽  
Multifractal Analysis ◽  
Point Processes ◽  
Multifractal Spectrum ◽  
Jump Processes ◽  
Poisson Point Processes ◽  
Infinite Products ◽  
Stationary Measures ◽  
Basic Properties ◽  
Side Result

There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced themultifractal products of stochastic processes(MPSP) and provided basic properties concerning convergence, nondegeneracy, and scaling of moments. This paper considers a subclass of MPSP which is determined by jump processes with i.i.d. exponentially distributed interjump times. Particularly, the information dimension and a multifractal spectrum of the MPSP are computed. As a side result it is shown that the random partitions imprinted naturally by a family of Poisson point processes are sufficient to determine the spectrum in this case.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

scholarly journals Extended and conditional versions of the PASTA property

1990 ◽  
Vol 22 (2) ◽  
pp. 510-512 ◽  
Author(s):  
Dieter König ◽  
Volker Schmidt
Keyword(s):  
Stochastic Processes ◽  
Point Processes ◽  
Long Run ◽  
Run Time

Two types of conditions are discussed ensuring the equality between long-run time fractions and long-run event fractions of stochastic processes with embedded point processes. Modifications of this equality statement are considered.

Analog Circuits Fault Diagnosis Using Multifractal Analysis

2013 ◽  
Vol 721 ◽  
pp. 367-371
Author(s):  
Yong Kui Sun ◽  
Zhi Bin Yu
Keyword(s):  
Support Vector Machine ◽  
Fault Diagnosis ◽  
Analog Circuits ◽  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Experimental Results ◽  
Support Vector ◽  
Diagnosis Accuracy ◽  
Circuit Under Test

Analog circuits fault diagnosis using multifractal analysis is presented in this paper. The faulty response of circuit under test is analyzed by multifratal formalism, and the fault feature consists of multifractal spectrum parameters. Support vector machine is used to identify the faults. Experimental results prove the proposed method is effective and the diagnosis accuracy reaches 98%.

Integral equations for compound distribution functions

1996 ◽  
Vol 33 (2) ◽  
pp. 388-399 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Point Processes ◽  
Marked Point ◽  
Building Blocks ◽  
Distribution Functions ◽  
Jump Processes ◽  
Marked Point Processes ◽  
Martingale Theory ◽  
Compound Distribution ◽  
Change Of Variable ◽  
Fixed Period

The aim of the present paper is to introduce some techniques, based on the change of variable formula for processes of finite variation, for establishing (integro) differential equations for evaluating the distribution of jump processes for a fixed period of time. This is of interest in insurance mathematics for evaluating the distribution of the total amount of claims occurred over some period of time, and attention will be given to such issues. Firstly we will study some techniques when the process has independent increments, and then a more refined martingale technique is discussed. The building blocks are delivered by the theory of marked point processes and associated martingale theory. A simple numerical example is given.

Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

2013 ◽  
pp. 16-46
Author(s):  
Oleg I. Sheluhin ◽  
Artem V. Garmashev
Keyword(s):  
Communication Systems ◽  
Multifractal Analysis ◽  
Fractal Dimensions ◽  
Multifractal Spectrum ◽  
Telecommunication Networks ◽  
Information Communication ◽  
Novel Approach ◽  
Wavelet Transform Modulus Maxima ◽  
Modulus Maxima ◽  
Development Methods

In this chapter, the main principles of the theory of fractals and multifractals are stated. A singularity spectrum is introduced for the random telecommunication traffic, concepts of fractal dimensions and scaling functions, and methods used in their determination by means of Wavelet Transform Modulus Maxima (WTMM) are proposed. Algorithm development methods for estimating multifractal spectrum are presented. A method based on multifractal data analysis at network layer level by means of WTMM is proposed for the detection of traffic anomalies in computer and telecommunication networks. The chapter also introduces WTMM as the informative indicator to exploit the distinction of fractal dimensions on various parts of a given dataset. A novel approach based on the use of multifractal spectrum parameters is proposed for estimating queuing performance for the generalized multifractal traffic on the input of a buffering device. It is shown that the multifractal character of traffic has significant impact on queuing performance characteristics.

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