Explicit Limits of Total Variation Distance in Approximations of Random Logarithmic Assemblies by Related Poisson Processes

1997 ◽  
Vol 6 (1) ◽  
pp. 87-105 ◽  
Author(s):  
DUDLEY STARK
Keyword(s):  
Total Variation ◽  
Local Limit ◽  
Total Variation Distance ◽  
Weighted Sums ◽  
Set Partitions ◽  
Ewens Sampling Formula ◽  
Combinatorial Objects ◽  
Sampling Formula ◽  
Random Mappings ◽  
Variation Distance

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i[ ]Zi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ε (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ε (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

Total Variation Asymptotics for Refined Poisson Process Approximations of Random Logarithmic Assemblies

1999 ◽  
Vol 8 (6) ◽  
pp. 567-598 ◽  
Author(s):  
DUDLEY STARK
Keyword(s):  
Poisson Process ◽  
Total Variation ◽  
Counting Process ◽  
Regular Graphs ◽  
Total Size ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Sampling Formula ◽  
Random Mappings ◽  
Variation Distance

Assemblies are decomposable combinatorial objects characterized by a sequence mi that counts the number of possible components of size i. Permutations on n elements, mappings from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are characterized by constants θ > 0 and κ0 > 0 such that miκi0/ (i−1)! → θ. Random mappings, permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.Given a logarithmic assembly, all representatives having total size n are chosen uniformly and a component counting process C(n) = (C1(n), C2(n), …, Cn(n)) is defined, where Ci(n) is the number of components of size i. Our results also apply to C(n) distributed as the Ewens sampling formula with parameter θ. Denote the component counting process up to size at most b by Cb(n) = (C1(n), C2(n), …, Cb(n)). It is natural to approximate Cb by Zb = (Z1, Z2, …, Zb), the b-dimensional process of independent Poisson variables Zi for which the ith variable has expectation [ ]Zi = miκi0 exp((1−θ)i/n)/i!. We find asymptotics for the total variation distance between Cb(n) and Zb.

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

scholarly journals Universality for Random Surfaces in Unconstrained Genus

10.37236/8623 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Thomas Budzinski ◽  
Nicolas Curien ◽  
Bram Petri
Keyword(s):  
Total Variation ◽  
Degree Sequence ◽  
Random Variable ◽  
Total Variation Distance ◽  
Alternative Proof ◽  
Arbitrary Sequence ◽  
Weak Version ◽  
Variation Distance ◽  
Planar Maps ◽  
Dirichlet Partition

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Convergence to stationary state for a Markov move-to-front scheme

1995 ◽  
Vol 32 (03) ◽  
pp. 768-776 ◽  
Author(s):  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Stationary State ◽  
Higher Order ◽  
Total Variation Distance ◽  
Variation Distance ◽  
Coupling Time

This work considers items (e.g. books, files) arranged in an array (e.g. shelf, tape) with N positions and assumes that items are requested according to a Markov chain (possibly, of higher order). After use, the requested item is returned to the leftmost position of the array. Successive applications of the procedure above give rise to a Markov chain on permutations. For equally likely items, the number of requests that makes this Markov chain close to its stationary state is estimated. To achieve that, a coupling argument and the total variation distance are used. Finally, for non-equally likely items and so-called p-correlated requests, the coupling time is presented as a function of the coupling time when requests are independent.

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