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.

scholarly journals Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space

2007 ◽  
Vol 44 (04) ◽  
pp. 1047-1055 ◽  
Author(s):  
K. J. E. Carpio ◽  
D. J. Daley
Keyword(s):  
State Space ◽  
Long Range ◽  
Transition Probability ◽  
Random Variable ◽  
Return Time ◽  
Long Range Dependence ◽  
Countable State Space ◽  
Countable State ◽  
Number Of Visits ◽  
Step Transition

When {X n } is an irreducible, stationary, aperiodic Markov chain on the countable state space X = {i, j,…}, the study of long-range dependence of any square integrable functional {Y n } := {y X n } of the chain, for any real-valued function {y i : i ∈ X }, involves in an essential manner the functions Q ij n = ∑ r=1 n (p ij r − π j ), where p ij r = P{X r = j | X 0 = i} is the r-step transition probability for the chain and {π i : i ∈ X } = P{X n = i} is the stationary distribution for {X n }. The simplest functional arises when Y n is the indicator sequence for visits to some particular state i, I ni = I {X n =i} say, in which case limsup n→∞ n −1var(Y 1 + ∙ ∙ ∙ + Y n ) = limsup n→∞ n −1 var(N i (0, n]) = ∞ if and only if the generic return time random variable T ii for the chain to return to state i starting from i has infinite second moment (here, N i (0, n] denotes the number of visits of X r to state i in the time epochs {1,…,n}). This condition is equivalent to Q ji n → ∞ for one (and then every) state j, or to E(T jj 2) = ∞ for one (and then every) state j, and when it holds, (Q ij n / π j ) / (Q kk n / π k ) → 1 for n → ∞ for any triplet of states i, j k.

scholarly journals Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space

2007 ◽  
Vol 44 (4) ◽  
pp. 1047-1055 ◽  
Author(s):  
K. J. E. Carpio ◽  
D. J. Daley
Keyword(s):  
State Space ◽  
Long Range ◽  
Transition Probability ◽  
Random Variable ◽  
Return Time ◽  
Long Range Dependence ◽  
Countable State Space ◽  
Countable State ◽  
Number Of Visits ◽  
Step Transition

When {Xn} is an irreducible, stationary, aperiodic Markov chain on the countable state space X = {i, j,…}, the study of long-range dependence of any square integrable functional {Yn} := {yXn} of the chain, for any real-valued function {yi: i ∈ X}, involves in an essential manner the functions Qijn = ∑r=1n(pijr − πj), where pijr = P{Xr = j | X0 = i} is the r-step transition probability for the chain and {πi: i ∈ X} = P{Xn = i} is the stationary distribution for {Xn}. The simplest functional arises when Yn is the indicator sequence for visits to some particular state i, Ini = I{Xn=i} say, in which case limsupn→∞n−1var(Y1 + ∙ ∙ ∙ + Yn) = limsupn→∞n−1 var(Ni(0, n]) = ∞ if and only if the generic return time random variable Tii for the chain to return to state i starting from i has infinite second moment (here, Ni(0, n] denotes the number of visits of Xr to state i in the time epochs {1,…,n}). This condition is equivalent to Qjin → ∞ for one (and then every) state j, or to E(Tjj2) = ∞ for one (and then every) state j, and when it holds, (Qijn / πj) / (Qkkn / πk) → 1 for n → ∞ for any triplet of states i, jk.

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