Large deviation principle in nonparametric estimation of marked point processes

AbstractWe study large deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large deviation principle for the rescaled process of these receivers as the connection threshold tends to 0. Finally, we show how these results can be used to develop importance sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

Download Full-text

Large Deviation Principle for Occupancy Problems With Colored Balls

10.21236/ada461905 ◽

2003 ◽

Author(s):

Paul Dupuis ◽

Carl Nuzman ◽

Phil Whiting

Keyword(s):

Large Deviation Principle ◽

Large Deviation ◽

Deviation Principle ◽

Occupancy Problems

Download Full-text

Marked point processes as limits of Markovian arrival streams

Journal of Applied Probability ◽

10.1017/s0021900200117371 ◽

1993 ◽

Vol 30 (02) ◽

pp. 365-372 ◽

Cited By ~ 19

Author(s):

Søren Asmussen ◽

Ger Koole

Keyword(s):

Point Process ◽

Point Processes ◽

Marked Point ◽

The State ◽

Marked Point Processes ◽

Marked Point Process ◽

State Transitions ◽

Poisson Arrival ◽

Convergence Of Moments ◽

Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Download Full-text

Large deviation principle for the field in prequantum classical statistical field theory

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025717500254 ◽

2017 ◽

Vol 20 (04) ◽

pp. 1750025

Author(s):

Andrei Khrennikov ◽

Achref Majid

Keyword(s):

Field Theory ◽

Random Field ◽

Field Model ◽

Large Deviation Principle ◽

Large Deviation ◽

Background Field ◽

Deviation Principle ◽

Statistical Field ◽

Statistical Field Theory

In this paper, we prove a large deviation principle for the background field in prequantum statistical field model. We show a number of examples by choosing a specific random field in our model.

Download Full-text

Perfect sampling for infinite server and loss systems

Advances in Applied Probability ◽

10.1017/s0001867800048825 ◽

2015 ◽

Vol 47 (03) ◽

pp. 761-786 ◽

Cited By ~ 4

Author(s):

Jose Blanchet ◽

Jing Dong

Keyword(s):

Point Processes ◽

Marked Point ◽

Heavy Traffic ◽

Arrival Rate ◽

Marked Point Processes ◽

Perfect Sampling ◽

Running Time ◽

Infinite Server ◽

Loss Systems ◽

Server System

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss systems. We use a variation of dominated coupling from the past. We first simulate a stationary infinite server system backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We then use the infinite server system as an upper bound process to simulate the loss system. The running time analysis of our perfect sampling algorithm for loss systems is performed in the quality-driven (QD) and the quality-and-efficiency-driven regimes. In both cases, we show that our algorithm achieves subexponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of complexity.

Download Full-text

Random Marked Sets

Advances in Applied Probability ◽

10.1239/aap/1346955256 ◽

2012 ◽

Vol 44 (3) ◽

pp. 603-616 ◽

Cited By ~ 10

Author(s):

F. Ballani ◽

Z. Kabluchko ◽

M. Schlather

Keyword(s):

Random Fields ◽

Covariance Function ◽

Point Processes ◽

Marked Point ◽

Positive Definiteness ◽

Gaussian Random Fields ◽

Marked Point Processes ◽

Random Set ◽

Geostatistical Methods ◽

New Class

We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in . Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

Download Full-text