scholarly journals Asymptotic behavior of dependence measures for Ornstein-Uhlenbeck model based on long memory processes

2021 ◽  
Author(s):  
Janusz Gajda ◽  
Agnieszka Wyłomańska
Keyword(s):  
Asymptotic Behavior ◽  
Long Range ◽  
Long Memory ◽  
Estimation Method ◽  
Stability Index ◽  
Infinite Variance ◽  
Memory Property ◽  
Starting Point ◽  
Alternative Measures ◽  
Dependence Measures

AbstractIn this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other L$$\acute{e}$$ e ´ vy process) by long range dependent processes based on $$\alpha -$$ α - stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for $$\alpha -$$ α - stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.

scholarly journals Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

2012 ◽  
Vol 44 (4) ◽  
pp. 1113-1141 ◽  
Author(s):  
Rafał Kulik ◽  
Philippe Soulier
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Volatility ◽  
Long Memory ◽  
Convergence Rates ◽  
Partial Sums ◽  
Infinite Variance ◽  
Tail Behavior ◽  
Stochastic Volatility Models ◽  
Crucial Difference ◽  
Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

scholarly journals Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

2012 ◽  
Vol 44 (04) ◽  
pp. 1113-1141
Author(s):  
Rafał Kulik ◽  
Philippe Soulier
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Volatility ◽  
Long Memory ◽  
Convergence Rates ◽  
Partial Sums ◽  
Infinite Variance ◽  
Tail Behavior ◽  
Stochastic Volatility Models ◽  
Crucial Difference ◽  
Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

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.

Power-law correlations, related models for long-range dependence and their simulation

2000 ◽  
Vol 37 (04) ◽  
pp. 1104-1109 ◽  
Author(s):  
Tilmann Gneiting
Keyword(s):  
Long Range ◽  
Power Law ◽  
Long Memory ◽  
Stationary Time Series ◽  
Long Range Dependence ◽  
Short Term ◽  
Permissible Range ◽  
Simultaneous Fitting ◽  
Straightforward Proof

Martin and Walker ((1997) J. Appl. Prob. 34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Dual long memory property in returns and volatility: Evidence from the CEE countries' stock markets

2009 ◽  
Vol 10 (2) ◽  
pp. 122-139 ◽  
Author(s):  
Adnan Kasman ◽  
Saadet Kasman ◽  
Erdost Torun
Keyword(s):  
Long Memory ◽  
Stock Markets ◽  
Memory Property

Estimation in Dynamic Linear Regression Models with Infinite Variance Errors

1993 ◽  
Vol 9 (4) ◽  
pp. 570-588 ◽  
Author(s):  
Keith Knight
Keyword(s):  
Asymptotic Behavior ◽  
Linear Regression ◽  
Regression Model ◽  
Linear Regression Model ◽  
Regression Models ◽  
Infinite Variance ◽  
Linear Regression Models ◽  
Asymptotically Normal ◽  
Second Moments ◽  
True Values

This paper considers the asymptotic behavior of M-estimates in a dynamic linear regression model where the errors have infinite second moments but the exogenous regressors satisfy the standard assumptions. It is shown that under certain conditions, the estimates of the parameters corresponding to the exogenous regressors are asymptotically normal and converge to the true values at the standard n−½ rate.

scholarly journals Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

2010 ◽  
Vol 42 (2) ◽  
pp. 509-527 ◽  
Author(s):  
Donata Puplinskaitė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Random Coefficient ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stable Law ◽  
Slope Coefficient ◽  
Finite Dimensional

Contemporaneous aggregation ofNindependent copies of a random-coefficient AR(1) process with random coefficienta∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalizationN1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficientahas probability density vanishing regularly ata= 1 with exponentb∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

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