scholarly journals Long range dependence of heavy-tailed random functions

2021 ◽  
Vol 58 (3) ◽  
pp. 569-593
Author(s):  
Rafal Kulik ◽  
Evgeny Spodarev
Keyword(s):  
Long Range ◽  
Random Function ◽  
Limit Theorems ◽  
Random Processes ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Random Functions ◽  
Heavy Tailed ◽  
Definition Of ◽  
Index Space

AbstractWe introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

scholarly journals On1/fNoise

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Signal Processing ◽  
Differential Equations ◽  
Random Function ◽  
Biomedical Signal Processing ◽  
Living Systems ◽  
Long Range Dependence ◽  
Biomedical Signal ◽  
Random Functions ◽  
The Third ◽  
Heavy Tailed

Due to the fact that1/fnoise gains the increasing interests in the field of biomedical signal processing and living systems, we present this introductive survey that may suffice to exhibit the elementary and the particularities of1/fnoise in comparison with conventional random functions. Three theorems are given for highlighting the particularities of1/fnoise. The first says that a random function with long-range dependence (LRD) is a1/fnoise. The secondindicates that a heavy-tailed random function is in the class of1/fnoise. The third provides a type of stochastic differential equations that produce1/fnoise.

scholarly journals Representations of hermite processes using local time of intersecting stationary stable regenerative sets

2020 ◽  
Vol 57 (4) ◽  
pp. 1234-1251
Author(s):  
Shuyang Bai
Keyword(s):  
Long Range ◽  
Local Time ◽  
Limit Theorems ◽  
Local Times ◽  
Long Range Dependence ◽  
Random Interval ◽  
Stationary Increments ◽  
Self Similar ◽  
Regenerative Sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

The nature of discrete second-order self-similarity

2003 ◽  
Vol 35 (02) ◽  
pp. 395-416 ◽  
Author(s):  
A. Gefferth ◽  
D. Veitch ◽  
I. Maricza ◽  
S. Molnár ◽  
I. Ruzsa
Keyword(s):  
Fixed Points ◽  
Long Range ◽  
Power Laws ◽  
Complete Classification ◽  
Second Order ◽  
General Definition ◽  
Long Range Dependence ◽  
Self Similarity ◽  
New Treatment ◽  
Definition Of

A new treatment of second-order self-similarity and asymptotic self-similarity for stationary discrete time series is given, based on the fixed points of a renormalisation operator with normalisation factors which are not assumed to be power laws. A complete classification of fixed points is provided, consisting of the fractional noise and one other class. A convenient variance time function approach to process characterisation is used to exhibit large explicit families of processes asymptotic to particular fixed points. A natural, general definition of discrete long-range dependence is provided and contrasted with common alternatives. The closely related discrete form of regular variation is defined, its main properties given, and its connection to discrete self-similarity explained. Folkloric results on long-range dependence are proved or disproved rigorously.

VECTOR RANDOM FIELDS WITH LONG-RANGE DEPENDENCE

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 249-258 ◽  
Author(s):  
CHUNSHENG MA
Keyword(s):  
Long Range ◽  
Random Fields ◽  
Random Function ◽  
Short Range ◽  
Diagonal Entry ◽  
Random Functions ◽  
Cross Covariance ◽  
Direct Covariance ◽  
Crucial Ingredient ◽  
Special Case

It is well-known that the crucial ingredient for a vector Gaussian random function is its covariance matrix, where a diagonal entry termed a direct covariance is simply the covariance function of a component but it seems no simple interpretation for an off-diagonal entry termed a cross covariance, which often make it hard to specify. In this paper we employ three approaches to derive vector random functions in space and/or time, which are not homogeneous (stationary) in general but contain the stationary case as a special case, and have long-range or short-range dependence.

scholarly journals Hurst Index of Functions of Long-Range-Dependent Markov Chains

2012 ◽  
Vol 49 (02) ◽  
pp. 451-471
Author(s):  
Barlas Oğuz ◽  
Venkat Anantharam
Keyword(s):  
Markov Chain ◽  
Long Range ◽  
Queueing Networks ◽  
Time Distribution ◽  
Return Time ◽  
Indicator Function ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Hurst Index ◽  
Positive Recurrent

A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

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