scholarly journals On the Ornstein–Zernike equation for stationary cluster processes and the random connection model

2017 ◽  
Vol 49 (4) ◽  
pp. 1260-1287 ◽  
Author(s):  
Günter Last ◽  
Sebastian Ziesche
Keyword(s):  
Point Process ◽  
Cluster Model ◽  
Combinatorial Properties ◽  
Palm Calculus ◽  
Fourier Theory ◽  
Associated Pair ◽  
Special Case ◽  
Random Connection Model ◽  
Palm Probabilities ◽  
Cluster Processes

Abstract In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

MARTINGALE CHARACTERIZATIONS OF INCREMENT PROCESSES IN A LOCALLY COMPACT GROUP

2003 ◽  
Vol 06 (04) ◽  
pp. 563-595 ◽  
Author(s):  
HERBERT HEYER ◽  
GYULA PAP
Keyword(s):  
Compact Group ◽  
Martingale Problem ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Fourier Theory ◽  
Group A ◽  
Tensor Square ◽  
Special Case

Martingale characterizations and the related martingale problem are studied for processes with independent (not necessarily stationary) increments in an arbitrary locally compact group. In the special case of a compact Lie group, a Lévy-type characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

scholarly journals Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

10.37236/2248 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tom Denton
Keyword(s):  
Schur Functions ◽  
Canonical Decomposition ◽  
Schur Function ◽  
Combinatorial Properties ◽  
Affine Permutations ◽  
New Perspective ◽  
Special Case ◽  
Arbitrary Affine

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam.  This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements.  We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation.  As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.

scholarly journals LIMIT DISTRIBUTIONS OF SOME STEREOLOGICAL ESTIMATORS IN WICKSELL'S CORPUSCLE PROBLEM

2011 ◽  
Vol 26 (2) ◽  
pp. 63 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Analytical Solution ◽  
Asymptotic Behaviour ◽  
Point Process ◽  
Cross Section ◽  
Homogeneous System ◽  
Spherical Particles ◽  
Limit Distributions ◽  
The Common ◽  
Special Case ◽  
Independent Identically Distributed

Suppose that a homogeneous system of spherical particles (d-spheres) with independent identically distributed radii is contained in some opaque d-dimensional body, and one is interested to estimate the common radius distribution. The only information one can get is by making a cross-section of that body with an s-flat (1 ≤ s ≤ d -1) and measuring the radii of the s-spheres and their midpoints. The analytical solution of (the hyper-stereological version of) Wicksell's corpuscle problem is used to construct an empirical radius distribution of the d-spheres. In this paper we study the asymptotic behaviour of this empirical radius distribution for s = d -1 and s = d - 2 under the assumption that the s-dimensional intersection volume becomes unboundedly large and the point process of the midpoints of the d-spheres is Brillinger-mixing. Of course, in stereological practice the only relevant cases are d = 3; s = 2 or s = 1 and d = 2; s = 1. Among others we generalize and extend some results obtained in Franklin (1981) and Groeneboom and Jongbloed (1995) under the Poisson assumption for the special case d = 3; s = 2.

Doubly stochastic Poisson processes and process control

1972 ◽  
Vol 4 (02) ◽  
pp. 318-338 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Point Process ◽  
Doubly Stochastic ◽  
Long Run ◽  
Finite State ◽  
Control State ◽  
Special Case ◽  
Finite State Space

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

A sample-path approach to Palm probabilities

1994 ◽  
Vol 31 (2) ◽  
pp. 430-437
Author(s):  
Shaler Stidham
Keyword(s):  
Path Analysis ◽  
Function Space ◽  
Point Process ◽  
Sample Path ◽  
Probability Measures ◽  
Sample Path Analysis ◽  
Inversion Formulas ◽  
One Dimensional ◽  
And Inversion ◽  
Palm Probabilities

Previous papers have established sample-path versions of relations between marginal time-stationary and event-stationary (Palm) state probabilities for a process with an imbedded point process. This paper extends the use of sample-path analysis to provide relations between frequencies for arbitrary (measurable) sets in function space, rather than just marginal (one-dimensional) frequencies. We define sample-path analogues of the time-stationary and event-stationary (Palm) probability measures for a process with an imbedded point process, and then derive sample-path versions of the Palm transformation and inversion formulas.

A sample-path approach to Palm probabilities

1994 ◽  
Vol 31 (02) ◽  
pp. 430-437
Author(s):  
Shaler Stidham
Keyword(s):  
Path Analysis ◽  
Function Space ◽  
Point Process ◽  
Sample Path ◽  
Probability Measures ◽  
Sample Path Analysis ◽  
Inversion Formulas ◽  
One Dimensional ◽  
And Inversion ◽  
Palm Probabilities

Previous papers have established sample-path versions of relations between marginal time-stationary and event-stationary (Palm) state probabilities for a process with an imbedded point process. This paper extends the use of sample-path analysis to provide relations between frequencies for arbitrary (measurable) sets in function space, rather than just marginal (one-dimensional) frequencies. We define sample-path analogues of the time-stationary and event-stationary (Palm) probability measures for a process with an imbedded point process, and then derive sample-path versions of the Palm transformation and inversion formulas.

scholarly journals Stationary partitions and Palm probabilities

2006 ◽  
Vol 38 (3) ◽  
pp. 602-620 ◽  
Author(s):  
Günter Last
Keyword(s):  
Probability Measure ◽  
Voronoi Tessellation ◽  
Random Partition ◽  
Weighted Version ◽  
Stationary Point Process ◽  
Typical Cell ◽  
Palm Measure ◽  
Special Case ◽  
Palm Probabilities ◽  
Shift Coupling

A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

scholarly journals A point process approach for spatial stochastic modeling of thunderstorm cells

2018 ◽  
Vol 37 (2) ◽  
pp. 472-496
Author(s):  
Bjoern Kriesche ◽  
Reinhold Hess ◽  
Volker Schmidt
Keyword(s):  
Point Process ◽  
Stochastic Modeling ◽  
Stochastic Geometry ◽  
Conditional Simulation ◽  
Process Approach ◽  
Model Parameters ◽  
Simulation Algorithm ◽  
Systematic Comparison ◽  
Doubly Stochastic ◽  
Cluster Processes

A POINT PROCESS APPROACH FOR SPATIAL STOCHASTIC MODELING OF THUNDERSTORM CELLSIn this paper we consider two different approaches for spatial stochastic modeling of thunderstorms. Thunderstorm cells are represented using germ-grain models from stochastic geometry, which are based on Coxor doubly-stochastic cluster processes. We present methods for the operationa lfitting of model parameters based on available point probabilities and thunderstorm records of past periods. Furthermore, we derive formulas forthe computation of point and area probabilities according to the proposed germ-grain models. We also introduce a conditional simulation algorithm in order to increase the model’s ability to precisely predict thunderstorm events. A systematic comparison of area probabilities, which are estimated from the proposed models, and thunderstorm records conclude the paper.

RENORMALIZATION OF COHERENT STATE VARIABLES, WITHIN THE GEOMETRICAL MAPPING OF ALGEBRAIC MODELS

2013 ◽  
Vol 22 (04) ◽  
pp. 1350022
Author(s):  
H. YÉPEZ-MARTÍNEZ ◽  
G. E. MORALES-HERNÁNDEZ ◽  
P. O. HESS ◽  
G. LÉVAI ◽  
P. R. FRASER
Keyword(s):  
Coherent State ◽  
Cluster Model ◽  
Coherent States ◽  
Quadrupole Deformation ◽  
State Variables ◽  
Algebraic Models ◽  
Interaction Terms ◽  
Special Case ◽  
Algebraic Counterpart

We investigate the geometrical mapping of algebraic models. As particular examples we consider the semimicriscopic algebraic cluster model (SACM) and the phenomenological algebraic cluster model (PACM), which also contains the vibron model as a special case. In the geometrical mapping, coherent states are employed as trial states. We show that the coherent state variables have to be renormalized and not the interaction terms of the Hamiltonian, as is usually done. The coherent state variables will depend on the total number of bosons and the coherent state variables. The nature of these variables is extracted through a relation obtained by comparing physical observables, such as the distance between the clusters or the quadrupole deformation of the nucleus, to their algebraic counterpart.

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