Random coefficient autoregression, regime switching and long memory

2003 ◽  
Vol 35 (03) ◽  
pp. 737-754 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Renewal Process ◽  
Long Memory ◽  
Regime Switching ◽  
Covariance Function ◽  
Random Coefficient ◽  
Levy Process ◽  
Partial Sums ◽  
Stable Lévy Process

We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Lévy process.

Random coefficient autoregression, regime switching and long memory

2003 ◽  
Vol 35 (3) ◽  
pp. 737-754 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Renewal Process ◽  
Long Memory ◽  
Regime Switching ◽  
Covariance Function ◽  
Random Coefficient ◽  
Levy Process ◽  
Partial Sums ◽  
Stable Lévy Process

We discuss long-memory properties and the partial sums process of the AR(1) process {Xt, t ∈ 𝕫} with random coefficient {at, t ∈ 𝕫} taking independent values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic Aj has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {Xt} decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {Xt} weakly converges to a stable Lévy process.

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).

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

2010 ◽  
Vol 42 (02) ◽  
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 of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 &lt; α &lt; 2) is discussed. We show that, under the normalization N 1/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 coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅ t } exhibits long memory. In particular, for {X̅ t }, 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).

AGGREGATION OF THE RANDOM COEFFICIENT GLARCH(1,1) PROCESS

2009 ◽  
Vol 26 (2) ◽  
pp. 406-425 ◽  
Author(s):  
Liudas Giraitis ◽  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Random Coefficient ◽  
Similar Process ◽  
Partial Sums ◽  
New Type ◽  
Self Similar

The paper discusses contemporaneous aggregation of the Linear ARCH (LARCH) model as defined in (1), which was introduced in Robinson (1991) and studied in Giraitis, Robinson, and Surgailis (2000) and other works. We show that the limiting aggregate of the (G)eneralized LARCH(1,1) process in (3)–(4) with random Beta distributed coefficient β exhibits long memory. In particular, we prove that squares of the limiting aggregated process have slowly decaying correlations and their partial sums converge to a self-similar process of a new type.

scholarly journals Quenched topological boundary modes can persist in a trivial system

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Ching Hua Lee ◽  
Justin C. W. Song
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Localized States ◽  
Topological Phase ◽  
Boundary Mode ◽  
Temporal Boundary ◽  
And Control ◽  
Topological Boundary

AbstractTopological boundary modes can occur at the spatial interface between a topological and gapped trivial phase and exhibit a wavefunction that exponentially decays in the gap. Here we argue that this intuition fails for a temporal boundary between a prequench topological phase that possess topological boundary eigenstates and a postquench gapped trivial phase that does not possess any eigenstates in its gap. In particular, we find that characteristics of states (e.g., probability density) prepared in a topologically non-trivial system can persist long after it is quenched into a gapped trivial phase with spatial profiles that appear frozen over long times postquench. After this near-stationary window, topological boundary mode profiles decay albeit, slowly in a power-law fashion. This behavior highlights the unusual features of nonequilibrium protocols enabling quenches to extend and control localized states of both topological and non-topological origins.

Area and volume quantification of arctic thaw slumps using time-series of digital elevation models generated from radar interferometry

2021 ◽  
Author(s):  
Philipp Bernhard ◽  
Simon Zwieback ◽  
Irena Hajnsek
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Volume Change ◽  
Digital Elevation Models ◽  
Probability Density Functions ◽  
The Arctic ◽  
Density Functions ◽  
Digital Elevation ◽  
Beta Coefficient ◽  
Area And Volume

&lt;p&gt;Vast areas of the Arctic host ice-rich permafrost, which is becoming increasingly vulnerable to terrain-altering thermokarst in a warming climate. Among the most rapid and dramatic changes are retrogressive thaw slumps. These slumps evolve by a retreat of the slump headwall during the summer months, making their change visible by comparing digital elevation models over time. In this study we use digital elevation models generated from single-pass radar TanDEM-X observations to derive volume and area change rates for retrogressive thaw slumps. At least three observations in the timespan from 2011 to 2017 are available with a spatial resolution of about 12 meter and a height sensitivity of about 0.5-2 meter. Our study regions include regions in Northern Canada (Peel Plateau/Richardson Mountains, Mackenzie River Delta Uplands, Ellesmere Island), Alaska (Noatak Valley) and Siberia (Yamal, Gydan, Taymyr, Chukotka) covering an area of 220.000 km&lt;sup&gt;2&lt;/sup&gt; with a total number of 1853 thaw slumps.&lt;/p&gt;&lt;p&gt;In this presentation we will focus on the area and volume change rate probability density functions of the mapped thaw slumps in these study areas. For landslides in temperate climate zones the area and volume change probability density function typically follow a distribution that can be characterized by three quantities: A rollover point defined as the peak in the distribution, a cutoff-point indicating the transition to a power law scaling for large landslides and the exponential beta coefficient of this power law. Here we will show that thaw slumps across the arctic follow indeed such a distribution and that the obtained values for the rollover, cutoff and beta coefficient can be used to distinguish between regions. Furthermore we will elaborate on possible reason why arctic thaw slumps can be described by such probability density functions as well as analyzing the differences between regions. This characterization can be useful to further improve our understanding of thaw slump initiation, the investigation of the drivers of their evolution as well as for modeling future thaw slump activity.&lt;/p&gt;

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.

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