Spectral covariance and limit theorems for random fields with infinite variance

Damarackas, Julius; Paulauskas, Vygantas

doi:10.1016/j.jmva.2016.09.013

Cited by 21 publications

(26 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Passing to random fields with infinite variance, we face with more difficult problems. First of all, measures of dependence for random fields in the case of infinite variance are less investigated in comparison with processes (for linear fields with stable innovations, preliminary results on the spectral covariance are presented in [28] and [8]; a more extensive study of this problem is in preparation). Secondly, the notion of memory, at least in everyday vocabulary, is naturally connected with the notion of the past (nobody speaks about memory when speaking about the future), and for processes with time as an argument, there are natural notions of the past and future, and therefore the notion of memory for processes is quite natural.…”

Section: Memory For Random Fieldsmentioning

confidence: 99%

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

Paulauskas

2016

Lith Math J

Self Cite

View full text Add to dashboard Cite

In the paper, we discuss how to define long, short, and negative memories for stationary processes on Z and fields on Z d with infinite variances. We propose to distinguish the properties of dependence and memory, and to attribute memory properties not only to a stationary random process (or a field) but also to a process and the operation that we apply to this process. We deal exclusively with the summation operation, that is, we consider the limit behavior of partial sums of random processes or fields. In order to have a unified approach to processes and fields with finite and infinite variances, we propose to define memory properties via the growth of normalizing sequences in limit theorems for partial sums. Also, we propose to change a little bit the terminology: instead of terms "long and short memories," to use positive and zero memories, respectively, leaving the term "negative memory" and introducing "strongly negative memory." For random fields, we introduce the notions of isotropic and directional memories.MSC: 60G10; 60G60

show abstract

Section: Memory For Random Fieldsmentioning

confidence: 99%

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

Paulauskas

2016

Lith Math J

Self Cite

View full text Add to dashboard Cite

show abstract

“…Regarding the SRD/LRD of the process X given in 2, our main result relies on the notion of

α

‐spectral covariance

ρ_{t} = \int_{double-struckR} {false(m false(prefix- x false) m false(t prefix- x false) false)}^{α false/ 2} d x

t \in double-struckR

, where m ( x ) ≥ 0,

x \in double-struckR

. The

α

‐spectral covariance was first introduced by Paulauskas (1976) and its properties were studied in Damarackas and Paulauskas (2014) and Damarackas and Paulauskas (2017). In Paulauskas (2016), it was discussed how the integrability of

ρ_{t}

can be used for the definition of the memory property.…”

Section: Introductionmentioning

confidence: 99%

Long range dependence for stable random processes

Makogin¹,

Oesting

Rapp³

et al. 2020

Journal Time Series Analysis

View full text Add to dashboard Cite

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 range dependence based on the covariance of indicators of excursion sets. Sufficient conditions for the long and short range dependence of α‐stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max‐stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence.

show abstract

“…Therefore, one can find alternative measures of dependence adequate to infinite variance processes. The most classical are covariation (and auto‐covariation) (Miller, ; Cambanis and Miller, ; Weron, ; Samorodnitsky and Taqqu, ; Gallagher, ; Nowicka‐Zagrajek and Wyłomańska, ), codifference (and auto‐codifference) (Nowicka‐Zagrajek and Wyłomańska, ; Rosadi, ; Rosadi and Deistler, ; Wyłomańska et al ) and fractional lower‐order covariance (Shao and Nikias, ; Ma and Nikias, ; Chen et al ), see also Damarackas and Paulauskas (). Furthermore, for the

α

‐stable‐based models the cross‐covariance and cross‐correlation (Mcleod, ; Kwang et al ; Genton and Kleiber, ) functions are not applicable to measure the relationship between components of the multi‐dimensional processes taking under consideration their delay in time.…”

Section: Introductionmentioning

confidence: 99%

Spatio‐Temporal Dependence Measures for Bivariate AR(1) Models with α‐Stable Noise

Grzesiek

Sikora

Teuerle

et al. 2019

Journal Time Series Analysis

View full text Add to dashboard Cite

Many real phenomena exhibit non‐Gaussian behavior. The non‐Gaussianity is manifested by impulsive behavior of the real data that can be found in both one‐dimensional and multi‐dimensional cases. Especially the multi‐dimensional datasets with non‐Gaussian behavior pose substantial analysis challenges to scientists and statisticians. In this article, we analyze the bidimensional vector autoregressive (VAR) model based on general bidimensional α‐stable distribution. This time series can be applied in modeling bidimensional data with impulsive behavior. We focus on the description of the spatio‐temporal dependence for analyzed bidimensional time series which in the considered case cannot be expressed in the language of the classical cross‐covariance or cross‐correlation function. We propose a new cross measure based on the alternative measure of dependence adequate for infinite variance processes, namely cross‐covariation. This article is an extension of the authors' previous work where the cross‐codifference was considered as the spatio‐temporal measure of the components of VAR model based on sub‐Gaussian distribution. In this article, we demonstrate that cross‐codifference and cross‐covariation can give different information about the relationships between components of bidimensional VAR models.

show abstract

Spectral covariance and limit theorems for random fields with infinite variance

Cited by 21 publications

References 29 publications

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

Long range dependence for stable random processes

Spatio‐Temporal Dependence Measures for Bivariate AR(1) Models with α‐Stable Noise

Contact Info

Product

Resources

About