# Long range dependence of heavy-tailed random functions

@article{Kulik2021LongRD, title={Long range dependence of heavy-tailed random functions}, author={Rafal Kulik and E. Spodarev}, journal={Journal of Applied Probability}, year={2021}, volume={58}, pages={569 - 593} }

Abstract We 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…

## 3 Citations

### Long range dependence for stable random processes

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We investigate long and short memory in α ‐stable moving averages and max‐stable processes with α ‐Fréchet marginal distributions. As these processes are heavy‐tailed, we rely on the notion of long…

### PR ] 2 9 A ug 2 01 9 Long Range Dependence for Stable Random Processes

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We investigate long and short memory in α-stable moving averages and max-stable processes with α-Fréchet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long…

### Detection of Long Range Dependence in the Time Domain for (In)Finite-Variance Time Series

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Empirical detection of long range dependence (LRD) of a time series often consists of deciding whether an estimate of the memory parameter d corresponds to LRD. Surprisingly, the literature oﬀers…

## References

SHOWING 1-10 OF 61 REFERENCES

### Long range dependence for stable random processes

- MathematicsJournal of Time Series Analysis
- 2020

We investigate long and short memory in α ‐stable moving averages and max‐stable processes with α ‐Fréchet marginal distributions. As these processes are heavy‐tailed, we rely on the notion of long…

### Nonstandard limit theorem for infinite variance functionals

- Mathematics
- 2008

We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a…

### A functional central limit theorem for integrals of stationary mixing random fields

- Mathematics
- 2015

We prove a functional central limit theorem for integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbb{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the…

### Extremal theory for long range dependent infinitely divisible processes

- MathematicsThe Annals of Probability
- 2019

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed…

### Central limit theorems for the excursion set volumes of weakly dependent random fields

- Mathematics
- 2012

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and…

### Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence

- Mathematics
- 2015

ABSTRACT The Karhunen–Loève expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian…

### Long memory random fields

- Mathematics
- 2006

A random field X = (Xn)n∈Zd is usually said to exhibit long memory, or strong dependence, or long-range dependence, when its covariance function r(n), n ∈ Z, is not absolutely summable : ∑ n∈Zd…

### Spectral covariance and limit theorems for random fields with infinite variance

- MathematicsJ. Multivar. Anal.
- 2017

### Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance

- Mathematics
- 1995

Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable…