scholarly journals Limit theorems for the fractional nonhomogeneous Poisson process

2019 ◽  
Vol 56 (01) ◽  
pp. 246-264 ◽  
Author(s):  
Nikolai Leonenko ◽  
Enrico Scalas ◽  
Mailan Trinh
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Poisson Process ◽  
Limit Theorems ◽  
Compound Poisson Process ◽  
Nonhomogeneous Poisson Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Compound Poisson ◽  
Finite Dimensional

AbstractThe 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.

scholarly journals Nonhomogeneous Poisson Process and Compound Poisson Process in the Modelling of Random Processes Related to Road Accidents

2019 ◽  
Vol 26 (1) ◽  
pp. 39-46
Author(s):  
Franciszek Grabski
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Statistical Distribution ◽  
Nonhomogeneous Poisson Process ◽  
Model Parameters ◽  
Time Interval ◽  
Road Accidents ◽  
Compound Poisson ◽  
The Road ◽  
The Given

Abstract The stochastic processes theory provides concepts and theorems that allow building probabilistic models concerning accidents. So called counting process can be applied for modelling the number of the road, sea and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The new theoretical results regarding compound Poisson process are presented in the paper. A nonhomogeneous Poisson process and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number and number of injured and killed people in the Polish road. To estimate model parameters were used data coming from the annual reports of the Polish police [9, 10]. Constructed models allowed anticipating number of accidents at any time interval with a length of h and the accident consequences. We obtained the expected value of fatalities or injured and the corresponding standard deviation in the given time interval. The statistical distribution of fatalities number in a single accident and statistical distribution of injured people number and also probability distribution of fatalities or injured number in a single accident are computed. It seems that the presented examples explain basic concepts and results discussed in the paper.

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (01) ◽  
pp. 23-41
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift. This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift.This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

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