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Thomas Michelitsch - One of the best experts on this subject based on the ideXlab platform.
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GENERALIZED FRACTIONAL Poisson Process AND RELATED STOCHASTIC DYNAMICS
Fractional Calculus and Applied Analysis, 2020Co-Authors: Thomas Michelitsch, Alejandro Pérez RiascosAbstract:We survey the 'generalized fractional Poisson Process' (GFPP). The GFPP is a renewal Process generalizing Laskin's fractional Poisson counting Process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prab-hakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson Process, the Erlang Process and the standard Poisson Process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems. MSC 2010 : Primary 60K05, 33E12, 26A33; Secondary 60J60, 65R10, 60K40
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Continuous time random walk and diffusion with generalized fractional Poisson Process
Physica A: Statistical Mechanics and its Applications, 2020Co-Authors: Thomas Michelitsch, Alejandro RiascosAbstract:A non-Markovian counting Process, the 'generalized fractional Poisson Process' (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters 0 < β ≤ 1, α > 0 and a time scale parameter. Generalizations to Laskin's fractional Poisson distribution and to the fractional Kolmogorov-Feller equation are derived. We develop a continuous time random walk subordinated to a GFPP in the infinite integer lattice Z d. For this stochastic motion, we deduce a 'generalized fractional diffusion equation'. In a well-scaled diffusion limit this motion is governed by the same type of fractional diffusion equation as with the fractional Poisson Process exhibiting subdiffusive t β-power law for the mean-square displacement. In the special cases α = 1 with 0 < β < 1 the equations of the Laskin fractional Poisson Process and for α = 1 with β = 1 the classical equations of the standard Poisson Process are recovered. The remarkably rich dynamics introduced by the GFPP opens a wide field of applications in anomalous transport and in the dynamics of complex systems.
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continuous time random walk and diffusion with generalized fractional Poisson Process
Physica A-statistical Mechanics and Its Applications, 2020Co-Authors: Thomas Michelitsch, Alejandro RiascosAbstract:Abstract A non-Markovian counting Process, the ‘generalized fractional Poisson Process’ (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters 0 β ≤ 1 , α > 0 and a time scale parameter. Generalizations to Laskin’s fractional Poisson distribution and to the fractional Kolmogorov–Feller equation are derived. We develop a continuous time random walk subordinated to a GFPP in the infinite integer lattice Z d . For this stochastic motion, we deduce a ‘generalized fractional diffusion equation’. For long observations, the generalized fractional diffusion exhibits the same power laws as fractional diffusion with fat-tailed waiting time densities and subdiffusive t β -power law for the expected number of arrivals. However, in short observation times, the GFPP exhibits distinct power-law patterns, namely t α β − 1 -scaling of the waiting time density and a t α β -pattern for the expected number of arrivals. The latter exhibits for α β > 1 superdiffusive behavior when the observation time is short. In the special cases α = 1 with 0 β 1 the equations of the Laskin fractional Poisson Process and for α = 1 with β = 1 the classical equations of the standard Poisson Process are recovered. The remarkably rich dynamics introduced by the GFPP opens a wide field of applications in anomalous transport and in the dynamics of complex systems.
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generalized fractional Poisson Process and related stochastic dynamics
arXiv: Statistical Mechanics, 2019Co-Authors: Thomas Michelitsch, Alejandro RiascosAbstract:We survey the 'generalized fractional Poisson Process' (GFPP). The GFPP is a renewal Process generalizing Laskin's fractional Poisson counting Process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges $0 0$ and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson Process, the Erlang Process and the standard Poisson Process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in $d$ dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.
Alejandro Pérez Riascos - One of the best experts on this subject based on the ideXlab platform.
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GENERALIZED FRACTIONAL Poisson Process AND RELATED STOCHASTIC DYNAMICS
Fractional Calculus and Applied Analysis, 2020Co-Authors: Thomas Michelitsch, Alejandro Pérez RiascosAbstract:We survey the 'generalized fractional Poisson Process' (GFPP). The GFPP is a renewal Process generalizing Laskin's fractional Poisson counting Process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prab-hakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson Process, the Erlang Process and the standard Poisson Process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems. MSC 2010 : Primary 60K05, 33E12, 26A33; Secondary 60J60, 65R10, 60K40
Mailan Trinh - One of the best experts on this subject based on the ideXlab platform.
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Limit theorems for the fractional nonhomogeneous Poisson Process
Journal of Applied Probability, 2019Co-Authors: Nikolai Leonenko, Enrico Scalas, Mailan TrinhAbstract:AbstractThe fractional nonhomogeneous Poisson Process was introduced by a time change of the nonhomogeneous Poisson Process with the inverseα-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson Process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson Process and the fractional compound Poisson Process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
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Limit theorems for the fractional non-homogeneous Poisson Process
Journal of Applied Probability, 2019Co-Authors: Nikolai Leonenko, Enrico Scalas, Mailan TrinhAbstract:The fractional nonhomogeneous Poisson Process was introduced by a time change of the nonhomogeneous Poisson Process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson Process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson Process and the fractional compound Poisson Process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
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the fractional non homogeneous Poisson Process
arXiv: Probability, 2016Co-Authors: Nikolai Leonenko, Enrico Scalas, Mailan TrinhAbstract:We introduce a non-homogeneous fractional Poisson Process by replacing the time variable in the fractional Poisson Process of renewal type with an appropriate function of time. We characterize the resulting Process by deriving its non-local governing equation. We further compute the first and second moments of the Process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from the specialization of the non-homogeneous Process.
Lawrence M. Leemis - One of the best experts on this subject based on the ideXlab platform.
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Nonparametric estimation of the Cumulative intensity function for a nonhomogeneous Poisson Process
Management Science, 1991Co-Authors: Lawrence M. LeemisAbstract:A nonparametric technique for estimating the cumulative intensity function of a nonhomogeneous Poisson Process from one or more realizations is developed. This technique does not require any arbitrary parameters from the modeler, and the estimated cumulative intensity function can be used to generate a point Process for Monte Carlo simulation by inversion. Three examples are given.
I Wayan Mangku - One of the best experts on this subject based on the ideXlab platform.
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CONSISTENCY OF A KERNEL-TYPE ESTIMATOR OF THE INTENSITY OF THE CYCLIC Poisson Process WITH THE LINEAR TREND
Journal of the Indonesian Mathematical Society, 2012Co-Authors: I Wayan Mangku, Siswadi, Retno BudiartiAbstract:A consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson Process in the presence of linear trend is constructed and investigated. It is assumed that only a single realization of the Poisson Process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window in definitely expands. DOI : http://dx.doi.org/10.22342/jims.15.1.42.37-48
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A note on estimation of the global intensity of a cyclic Poisson Process in the presence of linear trend
Journal of Mathematics and Its Applications, 2005Co-Authors: I Wayan MangkuAbstract:ABSTRACT. We construct and investigate a consistent kernel-type non parametric estimator of the global intensity of a cyclic Poisson Process in the presence of linear trend. It is assumed that only a single realization of the Poisson Process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands. The asymptotic bias and variance of the proposed estimator are computed. Bias reductionof the estimator is also proposed. 1991 Mathematics Subject Classification: 60G55, 62GO5, 62G20. Keywords and Phrases: cyclic Poisson Process, global intensity, linear trend, nonparametric estimation, consistency, bias, variance.
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ESTIMATING THE INTENSITY IN THE FORM OF A POWER FUNCTION OF AN INHOMOGENEOUS Poisson Process
Journal of Mathematics and Its Applications, 2005Co-Authors: I Wayan Mangku, I. Widiyastuti, I G. P. PurnabaAbstract:An estimator of the intensity in the form of a power function of an inhomogeneous Poisson Process is constructed and investigated. It is assumed that only a single realization of the Poisson Process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands. The asymptotic bias, variance and the mean- squared error of the proposed estimator are computed. Asymptotic normality of the estimator is also established.
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Consistent estimation of the intensity function of a cyclic Poisson Process
Journal of Multivariate Analysis, 2003Co-Authors: Roelof Helmers, I Wayan Mangku, Ričardas ZitikisAbstract:We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson Process when the period is unknown. We do not assume any particular parametric form for the intensity function, nor do we even assume that it is continuous. Moreover, we consider the situation when only a single realization of the Poisson Process is available, and only in a bounded window. We prove, in particular, that the proposed estimator is consistent when the size of the window indefinitely expands. We also obtain complete convergence of the estimator.
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Statistical properties of a kernel type estimator of the intensity function of a cyclic Poisson Process
2001Co-Authors: Roelof Helmers, I Wayan Mangku, Ričardas ZitikisAbstract:We consider a kernel-type nonparametric estimator of the intensity function of a cyclic Poisson Process when the period is unknown. We assume that only a single realization of the Poisson Process is observed in a bounded window which expands in time. We compute the asymptotic bias, variance, and the mean squared error of the estimator when the window indefinitely expands.
