Some simple properties of sums of random variables having long-range dependence

1989 ◽  
Vol 424 (1867) ◽  
pp. 255-262 ◽  
Keyword(s):  
Numerical Methods ◽  
Long Range ◽  
Random Variables ◽  
Higher Order ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Sums Of Random Variables ◽  
Tensor Methods ◽  
Non Gaussian ◽  
Special Case

The higher-order moments and cumulants of sums of a special case of random variables having long-range dependence are investigated. Tensor methods are used to simplify the calculations. The limiting form of the second and third cumulants as the number of variables added becomes large is studied by analytical and numerical methods. The implications are discussed for the existence of non-gaussian limits of sums of random quantities of finite variance and long-range dependence.

scholarly journals Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

2004 ◽  
Vol 41 (A) ◽  
pp. 35-53 ◽  
Author(s):  
V. V. Anh ◽  
N. N. Leonenko ◽  
L. M. Sakhno
Keyword(s):  
Long Range ◽  
Random Fields ◽  
Spectral Estimation ◽  
Kernel Estimation ◽  
Mathematical Finance ◽  
Higher Order ◽  
Long Range Dependence ◽  
Type Method ◽  
Minimum Contrast ◽  
Consistency And Asymptotic Normality

This paper provides a quasi-likelihood or minimum-contrast-type method for the parameter estimation of random fields in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general kth order and allows for possible long-range dependence in the random fields. To avoid bias due to edge effects, data tapering is incorporated into the method. The suggested minimum contrast functional is linear with respect to the periodogram of kth order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretization is not required in the estimation of continuously observed random fields. The consistency and asymptotic normality of the resulting estimators are established. Illustrative applications of the method to some problems in mathematical finance and signal detection 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 Estimation of marginal parameters of SUP-OU processes with long range dependence

2016 ◽  
Vol 4 (2) ◽  
pp. 102
Author(s):  
Emanuele Taufer
Keyword(s):  
Long Range ◽  
Asymptotic Theory ◽  
Marginal Distribution ◽  
Real Data ◽  
Function Technique ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Marginal Distributions ◽  
Non Gaussian ◽  
Basic Features

Superpositions of Ornstein Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non Gaussian marginal distributions. Estimation of the parameters of the marginal distribution is undertaken by means of a characteristic function technique. We provide the relevant asymptotic theory as well as results of simulations and real data applications.

scholarly journals Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

2004 ◽  
Vol 41 (A) ◽  
pp. 35-53
Author(s):  
V. V. Anh ◽  
N. N. Leonenko ◽  
L. M. Sakhno
Keyword(s):  
Long Range ◽  
Random Fields ◽  
Spectral Estimation ◽  
Kernel Estimation ◽  
Mathematical Finance ◽  
Higher Order ◽  
Long Range Dependence ◽  
Type Method ◽  
Minimum Contrast ◽  
Consistency And Asymptotic Normality

This paper provides a quasi-likelihood or minimum-contrast-type method for the parameter estimation of random fields in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general kth order and allows for possible long-range dependence in the random fields. To avoid bias due to edge effects, data tapering is incorporated into the method. The suggested minimum contrast functional is linear with respect to the periodogram of kth order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretization is not required in the estimation of continuously observed random fields. The consistency and asymptotic normality of the resulting estimators are established. Illustrative applications of the method to some problems in mathematical finance and signal detection are given.

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.

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