STIT tessellations are Bernoulli and standard

2012 ◽  
Vol 34 (3) ◽  
pp. 876-892 ◽  
Author(s):  
SERVET MARTÍNEZ
Keyword(s):  
Continuous Time ◽  
Bernoulli Shift ◽  
Random Sequence ◽  
Time Process ◽  
Continuous Time Process ◽  
Stit Tessellation ◽  
Bernoulli Flow

AbstractLet (Yt:t>0) be a STIT tessellation process and a>1. We prove that the random sequence (anYan:n∈ℤ) is a non-anticipating factor of a Bernoulli shift. We deduce that the continuous time process (atYat:t∈ℝ) is a Bernoulli flow. We use the techniques and results in Martínez and Nagel [Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134]. We also show that the filtration associated to the non-anticipating factor is standard in Vershik’s sense.

Stochastic ordering of random processes with an imbedded point process

1991 ◽  
Vol 28 (3) ◽  
pp. 553-567 ◽  
Author(s):  
François Baccelli
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Random Sequence ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Time Process ◽  
Continuous Time Process ◽  
Partial Orderings ◽  
Stationary Probabilities

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

Stochastic ordering of random processes with an imbedded point process

1991 ◽  
Vol 28 (03) ◽  
pp. 553-567 ◽  
Author(s):  
François Baccelli
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Random Sequence ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Time Process ◽  
Continuous Time Process ◽  
Partial Orderings ◽  
Stationary Probabilities

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

scholarly journals Mixed Spectra for Stable Signals from Discrete Observations

2021 ◽  
Vol 12 (05) ◽  
pp. 21-44
Author(s):  
Rachid Sabre
Keyword(s):  
Spectral Density ◽  
Continuous Time ◽  
Continuous Measurement ◽  
Spectral Measurement ◽  
Polynomial Kernel ◽  
Time Process ◽  
Discrete Observations ◽  
Continuous Part ◽  
Continuous Time Process ◽  
Discrete Measurement

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

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