On the rate of Poisson process approximation to a Bernoulli process

2004 ◽  
Vol 41 (01) ◽  
pp. 271-276
Author(s):  
P. S. Ruzankin
Keyword(s):  
Poisson Process ◽  
Total Variation ◽  
Point Process ◽  
Total Variation Distance ◽  
Bernoulli Process ◽  
Variation Distance ◽  
Kantorovich Distance ◽  
Process Approximation

The main result of the paper is a refinement of Xia's (1997) bound on the Kantorovich distance between distributions of a Bernoulli point process and an approximating Poisson process. In particular, we show that the distance between distributions of a Bernoulli point process and the Poisson process with the same mean measure has the order of the total variation distance between the laws of the total masses of these processes.

On the rate of Poisson process approximation to a Bernoulli process

2004 ◽  
Vol 41 (1) ◽  
pp. 271-276 ◽  
Author(s):  
P. S. Ruzankin
Keyword(s):  
Poisson Process ◽  
Total Variation ◽  
Point Process ◽  
Total Variation Distance ◽  
Bernoulli Process ◽  
Variation Distance ◽  
Kantorovich Distance ◽  
Process Approximation

The main result of the paper is a refinement of Xia's (1997) bound on the Kantorovich distance between distributions of a Bernoulli point process and an approximating Poisson process. In particular, we show that the distance between distributions of a Bernoulli point process and the Poisson process with the same mean measure has the order of the total variation distance between the laws of the total masses of these processes.

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

scholarly journals The Poisson–Dirichlet Distribution and the Scale-Invariant Poisson Process

1999 ◽  
Vol 8 (5) ◽  
pp. 407-416 ◽  
Author(s):  
RICHARD ARRATIA ◽  
A. D. BARBOUR ◽  
SIMON TAVARÉ
Keyword(s):  
Poisson Process ◽  
Explicit Formula ◽  
Total Variation ◽  
Dirichlet Process ◽  
Dirichlet Distribution ◽  
Total Variation Distance ◽  
Scale Invariant ◽  
Variation Distance ◽  
Distribution Of Points

We show that the Poisson–Dirichlet distribution is the distribution of points in a scale-invariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T[les ]1. Restricting both processes to (0, β] for 0<β[les ]1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.

scholarly journals Differential properties of semigroups and estimates of distances between stationary distributions of diffusions

2019 ◽  
Vol 485 (4) ◽  
pp. 399-404
Author(s):  
V. I. Bogachev ◽  
A. F. Miftakhov ◽  
S. V. Shaposhnikov
Keyword(s):  
Total Variation ◽  
Diffusion Processes ◽  
Total Variation Distance ◽  
Stationary Distributions ◽  
Variation Distance ◽  
Kantorovich Distance ◽  
Differential Properties

We obtain new bounds on the total variation distance and the Kantorovich distance between stationary distributions of diffusion processes in terms of certain distances between diffusion and drift coefficients.

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (04) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

scholarly journals Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mario Muñoz-Organero ◽  
Ramona Ruiz-Blázquez
Keyword(s):  
Total Variation ◽  
Traffic Congestion ◽  
Automatic Detection ◽  
Mass Function ◽  
Total Variation Distance ◽  
Statistical Moments ◽  
Probability Mass Function ◽  
Traffic Lights ◽  
Traffic Conditions ◽  
Variation Distance

The automatic detection of road related information using data from sensors while driving has many potential applications such as traffic congestion detection or automatic routable map generation. This paper focuses on the automatic detection of road elements based on GPS data from on-vehicle systems. A new algorithm is developed that uses the total variation distance instead of the statistical moments to improve the classification accuracy. The algorithm is validated for detecting traffic lights, roundabouts, and street-crossings in a real scenario and the obtained accuracy (0.75) improves the best results using previous approaches based on statistical moments based features (0.71). Each road element to be detected is characterized as a vector of speeds measured when a driver goes through it. We first eliminate the speed samples in congested traffic conditions which are not comparable with clear traffic conditions and would contaminate the dataset. Then, we calculate the probability mass function for the speed (in 1 m/s intervals) at each point. The total variation distance is then used to find the similarity among different points of interest (which can contain a similar road element or a different one). Finally, a k-NN approach is used for assigning a class to each unlabelled element.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

scholarly journals Extremum Problems With Total Variation Distance and Their Applications

2014 ◽  
Vol 59 (9) ◽  
pp. 2353-2368 ◽  
Author(s):  
Charalambos D. Charalambous ◽  
Ioannis Tzortzis ◽  
Sergey Loyka ◽  
Themistoklis Charalambous
Keyword(s):  
Total Variation ◽  
Total Variation Distance ◽  
Extremum Problems ◽  
Variation Distance
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