Joint exceedances of high levels under a local dependence condition

1993 ◽  
Vol 30 (01) ◽  
pp. 112-120
Author(s):  
Helena Ferreira
Keyword(s):  
Order Statistics ◽  
Long Range ◽  
Joint Distribution ◽  
Long Range Dependence ◽  
Local Dependence ◽  
Stationary Sequences ◽  
Compound Poisson ◽  
Dependence Conditions

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.

Joint exceedances of high levels under a local dependence condition

1993 ◽  
Vol 30 (1) ◽  
pp. 112-120 ◽  
Author(s):  
Helena Ferreira
Keyword(s):  
Order Statistics ◽  
Long Range ◽  
Joint Distribution ◽  
Long Range Dependence ◽  
Local Dependence ◽  
Stationary Sequences ◽  
Compound Poisson ◽  
Dependence Conditions

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.

On the compound Poisson limit for high level exceedances

1989 ◽  
Vol 26 (03) ◽  
pp. 458-465 ◽  
Author(s):  
Jonathan Cohen
Keyword(s):  
Joint Distribution ◽  
Multiplicity Distribution ◽  
Claim Data ◽  
Stationary Sequence ◽  
Long Range Dependence ◽  
Insurance Claim ◽  
Compound Poisson ◽  
Insurance Claim Data ◽  
Poisson Limit ◽  
High Level

Let ∊ 1 ∊ 2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un (τ)) of [3] for every τ > 0, where nP(∊ 1 > un (τ))→ τ . Assume only that P (there are j exceedances of un (τ) by ∊ 1, ∊ 2, · ··, ∊ n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.

VALID EDGEWORTH EXPANSION FOR THE SAMPLE AUTOCORRELATION FUNCTION UNDER LONG RANGE DEPENDENCE

2001 ◽  
Vol 17 (1) ◽  
pp. 257-275 ◽  
Author(s):  
Offer Lieberman ◽  
Judith Rousseau ◽  
David M. Zucker
Keyword(s):  
Long Range ◽  
Autocorrelation Function ◽  
Simulation Study ◽  
Long Memory ◽  
Joint Distribution ◽  
Edgeworth Expansion ◽  
Long Range Dependence ◽  
Memory Process ◽  
Long Memory Process

We prove in this paper the validity of an Edgeworth expansion to the joint distribution of the sample autocorrelations of a stationary Gaussian long memory process. The method of proof relies on a verification of the suitably modified conditions for the validity of a multivariate Edgeworth expansion of Durbin (1980, Biometrika 67, 311–333). A simulation study proves the expansion to be useful and accurate.

scholarly journals Extremal clustering in non-stationary random sequences

Extremes ◽  
2021 ◽  
Author(s):  
Graeme Auld ◽  
Ioannis Papastathopoulos
Keyword(s):  
Asymptotic Distribution ◽  
Long Range ◽  
Extreme Values ◽  
Random Variables ◽  
Limiting Distribution ◽  
Long Range Dependence ◽  
Dependence Structure ◽  
Stationary Sequences ◽  
Random Sequences ◽  
Extremal Clustering

AbstractIt is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.

On the compound Poisson limit for high level exceedances

1989 ◽  
Vol 26 (3) ◽  
pp. 458-465 ◽  
Author(s):  
Jonathan Cohen
Keyword(s):  
Joint Distribution ◽  
Multiplicity Distribution ◽  
Claim Data ◽  
Stationary Sequence ◽  
Long Range Dependence ◽  
Insurance Claim ◽  
Compound Poisson ◽  
Insurance Claim Data ◽  
Poisson Limit ◽  
High Level

Let ∊1∊2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un(τ)) of [3] for every τ > 0, where nP(∊1> un(τ))→ τ. Assume only that P (there are j exceedances of un(τ) by ∊1, ∊2, · ··, ∊n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.

Central limit theorems for nearly long range dependent subordinated linear processes

2020 ◽  
Vol 57 (2) ◽  
pp. 637-656
Author(s):  
Martin Wendler ◽  
Wei Biao Wu
Keyword(s):  
Long Range ◽  
Central Limit ◽  
Short Range ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Arbitrary Power

AbstractThe limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

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.

scholarly journals On nonparametric regression for bivariate circular long-memory time series

2021 ◽  
Author(s):  
Jan Beran ◽  
Britta Steffens ◽  
Sucharita Ghosh
Keyword(s):  
Time Series ◽  
Nonparametric Regression ◽  
Long Range ◽  
Long Memory ◽  
Regression Function ◽  
Confidence Bands ◽  
Long Range Dependence ◽  
Asymptotic Results ◽  
Memory Time ◽  
Circular Time Series

AbstractWe consider nonparametric regression for bivariate circular time series with long-range dependence. Asymptotic results for circular Nadaraya–Watson estimators are derived. Due to long-range dependence, a range of asymptotically optimal bandwidths can be found where the asymptotic rate of convergence does not depend on the bandwidth. The result can be used for obtaining simple confidence bands for the regression function. The method is illustrated by an application to wind direction data.

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