scholarly journals Concave Majorants of Random Walks and Related Poisson Processes

2011 ◽  
Vol 20 (5) ◽  
pp. 651-682
Author(s):  
JOSH ABRAMSON ◽  
JIM PITMAN
Keyword(s):  
Random Walks ◽  
Point Process ◽  
Finite Length ◽  
Poisson Point Process ◽  
Complete Information ◽  
Arithmetic Mean ◽  
Poisson Processes ◽  
Infinite Length ◽  
Unified Approach ◽  
Geometric Length

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

scholarly journals Persistence and Equilibria of Branching Populations with Exponential Intensity

2012 ◽  
Vol 49 (1) ◽  
pp. 226-244
Author(s):  
Zakhar Kabluchko
Keyword(s):  
Laplace Transform ◽  
Random Walks ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Local Extinction ◽  
Branching Random Walks ◽  
Cluster Distribution

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form eλ(du) = e-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλχ with intensity eλ. If λφ'(λ) < 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Πceλχ over c > 0 and λ ∈ Kst, where Kst = {λ ∈ R: φ(λ) = 0, λφ'(λ) > 0}.

scholarly journals POISSON PROCESSES FOR SUBSYSTEMS OF FINITE TYPE IN SYMBOLIC DYNAMICS

2009 ◽  
Vol 09 (03) ◽  
pp. 393-422 ◽  
Author(s):  
JEAN-RENÉ CHAZOTTES ◽  
ZAQUEU COELHO ◽  
PIERRE COLLET
Keyword(s):  
Finite Type ◽  
Point Process ◽  
Symbolic Dynamics ◽  
Poisson Point Process ◽  
Dirac Point ◽  
Poisson Processes ◽  
Constant Parameter ◽  
Shift Of Finite Type ◽  
The Times ◽  
Defining Graph

Let Δ ⊊ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type [Formula: see text]. Let ΣΔ be the subshift of allowable paths in the graph of [Formula: see text] which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on [Formula: see text], let τn be the point process defined as the sum of Dirac point masses at the times k > 0, suitably rescaled, for which the first n-symbols of Tkx belong to Δ. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of φ to ΣΔ and the parameters of the limit law are explicitly computed.

Fluctuation identities for lévy processes and splitting at the maximum

1980 ◽  
Vol 12 (4) ◽  
pp. 893-902 ◽  
Author(s):  
Priscilla Greenwood ◽  
Jim Pitman
Keyword(s):  
Point Process ◽  
Lévy Processes ◽  
Poisson Point Process ◽  
Levy Processes ◽  
Fluctuation Theory ◽  
Unified Approach ◽  
Path Decomposition

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.

scholarly journals Persistence and Equilibria of Branching Populations with Exponential Intensity

2012 ◽  
Vol 49 (01) ◽  
pp. 226-244
Author(s):  
Zakhar Kabluchko
Keyword(s):  
Laplace Transform ◽  
Random Walks ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Local Extinction ◽  
Branching Random Walks ◽  
Cluster Distribution

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form e λ(du) = e-λu du, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) &gt; 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλ χ with intensity e λ. If λφ'(λ) &lt; 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Π ce λ χ over c &gt; 0 and λ ∈ K st, where K st = {λ ∈ R: φ(λ) = 0, λφ'(λ) &gt; 0}.

Fluctuation identities for lévy processes and splitting at the maximum

1980 ◽  
Vol 12 (04) ◽  
pp. 893-902 ◽  
Author(s):  
Priscilla Greenwood ◽  
Jim Pitman
Keyword(s):  
Point Process ◽  
Lévy Processes ◽  
Poisson Point Process ◽  
Levy Processes ◽  
Fluctuation Theory ◽  
Unified Approach ◽  
Path Decomposition

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.

scholarly journals Extremes of multitype branching random walks: heaviest tail wins

2019 ◽  
Vol 51 (2) ◽  
pp. 514-540
Author(s):  
Ayan Bhattacharya ◽  
Krishanu Maulik ◽  
Zbigniew Palmowski ◽  
Parthanil Roy
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Weak Limit ◽  
The Other ◽  
Branching Random Walk ◽  
Branching Random Walks

AbstractWe consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the nth generation.

Shear-Driven Flow in a Toroid of Square Cross Section

2003 ◽  
Vol 125 (1) ◽  
pp. 130-137 ◽  
Author(s):  
J. A. C. Humphrey ◽  
J. Cushner ◽  
M. Al-Shannag ◽  
J. Herrero ◽  
F. Giralt
Keyword(s):  
Cross Section ◽  
Finite Length ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Transition To Turbulence ◽  
Radius Of Curvature ◽  
Infinite Length ◽  
Core Flow ◽  
Reynolds Number Flow ◽  
End Wall

The two-dimensional wall-driven flow in a plane rectangular enclosure and the three-dimensional wall-driven flow in a parallelepiped of infinite length are limiting cases of the more general shear-driven flow that can be realized experimentally and modeled numerically in a toroid of rectangular cross section. Present visualization observations and numerical calculations of the shear-driven flow in a toroid of square cross section of characteristic side length D and radius of curvature Rc reveal many of the features displayed by sheared fluids in plane enclosures and in parallelepipeds of infinite as well as finite length. These include: the recirculating core flow and its associated counterrotating corner eddies; above a critical value of the Reynolds (or corresponding Goertler) number, the appearance of Goertler vortices aligned with the recirculating core flow; at higher values of the Reynolds number, flow unsteadiness, and vortex meandering as precursors to more disorganized forms of motion and eventual transition to turbulence. Present calculations also show that, for any fixed location in a toroid, the Goertler vortex passing through that location can alternate its sense of rotation periodically as a function of time, and that this alternation in sign of rotation occurs simultaneously for all the vortices in a toroid. This phenomenon has not been previously reported and, apparently, has not been observed for the wall-driven flow in a finite-length parallelepiped where the sense of rotation of the Goertler vortices is determined and stabilized by the end wall vortices. Unlike the wall-driven flow in a finite-length parallelepiped, the shear-driven flow in a toroid is devoid of contaminating end wall effects. For this reason, and because the toroid geometry allows a continuous variation of the curvature parameter, δ=D/Rc, this flow configuration represents a more general paradigm for fluid mechanics research.

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