DEPENDENCE STRUCTURE OF A RENEWAL-REWARD PROCESS WITH INFINITE VARIANCE

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 185-192 ◽  
Author(s):  
JOSHUA B. LEVY ◽  
MURAD S. TAQQU
Keyword(s):  
Long Range ◽  
Power Function ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Dependence Structure ◽  
Reward Process ◽  
Renewal Reward Process ◽  
Stable Noise

The on-off renewal-reward process used to explain long-range dependence in Ethernet traffic can be extended to the case where, not only the inter-renewal times but also the rewards have infinite variance. The covariation and the codifference, which generalize the covariance to the infinite variance case, are computed for the limiting process. It is shown that they decay like a power function. The exponent of that power is the same as for fractional stable noise, even though the increments of the limiting process are different from fractional stable noise.

Parametric modelling of turbulence

1990 ◽  
Vol 332 (1627) ◽  
pp. 439-455 ◽  
Keyword(s):  
Long Range ◽  
Three Dimensional ◽  
Large Data ◽  
Theoretical Work ◽  
Long Range Dependence ◽  
Dependence Structure ◽  
Data Sets ◽  
Parametric Modelling ◽  
First Order ◽  
Log Linear

Some steps are taken towards a parametric statistical model for the velocity and velocity derivative fields in stationary turbulence, building on the background of existing theoretical and empirical knowledge of such fields. While the ultimate goal is a model for the three-dimensional velocity components, and hence for the corresponding velocity derivatives, we concentrate here on the stream wise velocity component. Discrete and continuous time stochastic processes of the first-order autoregressive type and with one-dimensional marginals having log-linear tails are constructed and compared with two large data-sets. It turns out that a first-order autoregression that fits the local correlation structure well is not capable of describing the correlations over longer ranges. A good fit locally as well as at longer ranges is achieved by using a process that is the sum of two independent autoregressions. We study this type of model in some detail. We also consider a model derived from the above-mentioned autoregressions and with dependence structure on the borderline to long-range dependence. This model is obtained by means of a general method for construction of processes with long-range dependence. Some suggestions for future empirical and theoretical work are given.

APPLYING BUCKET RANDOM PERMUTATIONS TO STATIONARY SEQUENCES WITH LONG-RANGE DEPENDENCE

Fractals ◽  
2007 ◽  
Vol 15 (02) ◽  
pp. 105-126 ◽  
Author(s):  
YINGCHUN ZHOU ◽  
MURAD S. TAQQU
Keyword(s):  
Long Range ◽  
Short Range ◽  
Random Permutation ◽  
Stationary Sequence ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Dependence Structure ◽  
Random Permutations ◽  
Medium Range ◽  
The Time Domain

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

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.

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

2012 ◽  
Vol 49 (2) ◽  
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.

scholarly journals A long range dependence stable process and an infinite variance branching system

2007 ◽  
Vol 35 (2) ◽  
pp. 500-527 ◽  
Author(s):  
Tomasz Bojdecki ◽  
Luis G. Gorostiza ◽  
Anna Talarczyk
Keyword(s):  
Long Range ◽  
Stable Process ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Branching System

On defining long-range dependence

1997 ◽  
Vol 34 (04) ◽  
pp. 939-944 ◽  
Author(s):  
C. C. Heyde ◽  
Y. Yang
Keyword(s):  
Long Range ◽  
Second Order ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stationary Processes ◽  
Domain Of Applicability

Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.

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.

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