Fractional Poisson process: long-range dependence and applications in ruin theory - Correction

2016 ◽  
Vol 53 (4) ◽  
pp. 1271-1272 ◽  
Author(s):  
R. Biard ◽  
B. Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Long Range Dependence ◽  
Ruin Theory ◽  
Fractional Poisson Process

scholarly journals Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

2014 ◽  
Vol 51 (3) ◽  
pp. 727-740 ◽  
Author(s):  
Romain Biard ◽  
Bruno Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Risk Model ◽  
Nonstationary Process ◽  
Long Range Dependence ◽  
Insurance Company ◽  
Surplus Process ◽  
Fractional Poisson Process ◽  
Renewal Risk ◽  
Dependence Property

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

scholarly journals On the long-range dependence of fractional Poisson and negative binomial processes

2016 ◽  
Vol 53 (4) ◽  
pp. 989-1000 ◽  
Author(s):  
A. Maheshwari ◽  
P. Vellaisamy
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Short Range ◽  
Negative Binomial ◽  
Stationary Process ◽  
Long Range Dependence ◽  
Fractional Poisson Process ◽  
Poissonian Noise ◽  
Negative Binomial Process ◽  
Binomial Process

Abstract We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

scholarly journals Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

2014 ◽  
Vol 51 (03) ◽  
pp. 727-740 ◽  
Author(s):  
Romain Biard ◽  
Bruno Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Risk Model ◽  
Nonstationary Process ◽  
Long Range Dependence ◽  
Insurance Company ◽  
Surplus Process ◽  
Fractional Poisson Process ◽  
Renewal Risk ◽  
Dependence Property

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

scholarly journals Fluctuations and pseudo long range dependence in network flows: A non-stationary Poisson process model

2009 ◽  
Vol 18 (4) ◽  
pp. 1373-1379 ◽  
Author(s):  
Chen Yu-Dong ◽  
Li Li ◽  
Zhang Yi ◽  
Hu Jian-Ming
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Process Model ◽  
Network Flows ◽  
Long Range Dependence

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe 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 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.

scholarly journals Representations of hermite processes using local time of intersecting stationary stable regenerative sets

2020 ◽  
Vol 57 (4) ◽  
pp. 1234-1251
Author(s):  
Shuyang Bai
Keyword(s):  
Long Range ◽  
Local Time ◽  
Limit Theorems ◽  
Local Times ◽  
Long Range Dependence ◽  
Random Interval ◽  
Stationary Increments ◽  
Self Similar ◽  
Regenerative Sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

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.

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