scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (2) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (02) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (04) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes. The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

High-Level exceedances of regenerative and semi-stationary processes

1980 ◽  
Vol 17 (02) ◽  
pp. 423-431 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Point Processes ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Partial Sums ◽  
Stationary Processes ◽  
Special Cases ◽  
Cumulative Amount ◽  
High Level ◽  
Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

scholarly journals Lévy-based Cox point processes

2008 ◽  
Vol 40 (3) ◽  
pp. 603-629 ◽  
Author(s):  
Gunnar Hellmund ◽  
Michaela Prokešová ◽  
Eva B. Vedel Jensen
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Kernel Smoothing ◽  
Random Measure ◽  
Infinitely Divisible ◽  
Mixing Properties ◽  
Cox Processes ◽  
Special Cases ◽  
Spatio Temporal ◽  
Product Densities

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (04) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (4) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

High-Level exceedances of regenerative and semi-stationary processes

1980 ◽  
Vol 17 (2) ◽  
pp. 423-431 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Point Processes ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Partial Sums ◽  
Stationary Processes ◽  
Special Cases ◽  
Cumulative Amount ◽  
High Level ◽  
Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

scholarly journals Lévy-based Cox point processes

2008 ◽  
Vol 40 (03) ◽  
pp. 603-629 ◽  
Author(s):  
Gunnar Hellmund ◽  
Michaela Prokešová ◽  
Eva B. Vedel Jensen
Keyword(s):  
Shot Noise ◽  
Point Processes ◽  
Kernel Smoothing ◽  
Random Measure ◽  
Infinitely Divisible ◽  
Mixing Properties ◽  
Cox Processes ◽  
Special Cases ◽  
Spatio Temporal ◽  
Product Densities

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

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