On multiple returns in the random-walk problem

1954 ◽  
Vol 50 (4) ◽  
pp. 586-591 ◽  
Author(s):  
C. Domb
Keyword(s):  
Random Walk ◽  
Lattice Point ◽  
Two Dimensions ◽  
Lattice Points ◽  
Detailed Calculation ◽  
Contour Integrals ◽  
Simple Lattice ◽  
Starting Point

Consider a random-walk problem on a simple lattice, the probabilities of the walker taking any direction in the lattice at each lattice point being equal. Then Polya (6) has shown that if a walker starts at the origin and continues to walk indefinitely, the probability of his passing through his starting point is unity in one and two dimensions, but less than unity in three or more dimensions. Recently, a generalization of this problem has been considered (1),(3) in which the walker is allowed to jump several lattice points with assigned probabilities. F. G. Foster and I. J. Good (3) have shown that if the assigned probabilities satisfy certain conditions, Polya's result still holds, and K. L. Chung and W. H. J. Fuchs (1) have shown that the result is valid under far less restrictive conditions. The above authors were primarily concerned with the question whether return is almost certain or not, and did not consider a detailed calculation of the probability at any stage. It is the purpose of the present paper to show that the use of contour integrals allied with the method of steepest descents (4) enables one to perform this calculation very simply.

NULL RECURRENT UNIT ROOT PROCESSES

2011 ◽  
Vol 28 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Terje Myklebust ◽  
Hans Arnfinn Karlsen ◽  
Dag Tjøstheim
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Unit Circle ◽  
Unit Root ◽  
Markov Models ◽  
Two Dimensions ◽  
Property A ◽  
Starting Point ◽  
Recurrent Markov Chain ◽  
Null Recurrence

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and β-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is β-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made β-null recurrent for any β ∈ (0, 1) and can also be made null recurrent without being β-null recurrent.

Some explicit results for an asymmetric two-dimensional random walk

1963 ◽  
Vol 59 (2) ◽  
pp. 451-462 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Lattice Point ◽  
General Interest ◽  
Two Dimensional ◽  
General Lattice ◽  
Methods Of Analysis ◽  
Starting Point ◽  
Accessible Point ◽  
Absorption Probabilities

AbstractThree distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

Fans and Brands

2018 ◽  
pp. 18-34 ◽  
Author(s):  
Breanna M. Todd ◽  
Catherine A. Armstrong Soule
Keyword(s):  
Social Status ◽  
Pop Culture ◽  
Two Dimensions ◽  
Brand Community ◽  
Community Based ◽  
Relationship Type ◽  
Brand Communities ◽  
Starting Point ◽  
Rich History

Although fandom has a rich history within pop culture, it is difficult to know when fandom was formed and what constitutes a fandom. In this chapter the authors define fandom and its related activities, as well as delineate it from the similar fan-brand communities of brand communities and brand publics. A typology for fan-brand communities is presented with two dimensions: 1) motivation for engagement and 2) social status and relationship type. This typology can help guide researchers, brands, and marketers in effectively managing different subcultures of fans. This chapter may be used as a starting point for further understanding of fan-brand community-based relationships.

Brands, Fans, and Exchanges

2022 ◽  
pp. 1-19
Author(s):  
Breanna M. Todd ◽  
Catherine Anne Armstrong Soule
Keyword(s):  
Social Status ◽  
Pop Culture ◽  
Two Dimensions ◽  
Brand Community ◽  
Community Based ◽  
Relationship Type ◽  
Brand Communities ◽  
Starting Point ◽  
Different Types ◽  
Rich History

Although fandom has a rich history within pop culture, it is difficult to know what constitutes a fandom, what differentiates fandoms from similar phenomena as well as what different types of fandoms exist and how fandoms are formed and maintained. In this chapter, the authors define fandom and the related member actions that create and maintain fandoms, as well as delineate the concept from the similar fan-brand communities of transactional brand communities, social brand communities and brand publics. A typology for fan-brand communities is presented with two dimensions: 1) motivation for engagement; and 2) social status and relationship type. This typology can help guide researchers, brands, and marketers in effectively managing different subcultures of fans. This chapter may be used as a starting point for further understanding of fan-brand community-based relationships.

scholarly journals Wind Accretion by Compact Objects: The “Flip-Flop” Instability

1992 ◽  
Vol 151 ◽  
pp. 185-194
Author(s):  
Mario Livio
Keyword(s):  
Random Walk ◽  
Accretion Rate ◽  
Compact Object ◽  
Two Dimensions ◽  
Compact Objects ◽  
Numerical Calculations ◽  
Three Dimensions ◽  
Flip Flop ◽  
Accretion Flows ◽  
The Stability

The problem of the stability of wind accretion onto compact objects is examined. Recent analytical and numerical calculations show that in two dimensions, Bondi-Hoyle accretion flows are unstable to a “flip-flop” instability. The instability can manifest itself as bursts in the accretion rate and as a random walk-type spin-up, spin-down behaviour of the accreting compact object. The nature of the flow in three dimensions needs further clarification. Possible observational implications are reviewed.

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

Theoretical Investigation of Alloy Phase Equilibria by Continuous Displacement Cluster Variation Method

2011 ◽  
Vol 172-174 ◽  
pp. 1119-1127
Author(s):  
Tetsuo Mohri
Keyword(s):  
Phase Equilibria ◽  
Lattice Point ◽  
Cluster Variation Method ◽  
Lattice Points ◽  
Variation Method ◽  
Bravais Lattice ◽  
Diffuse Intensity ◽  
Wide Range ◽  
Cluster Variation ◽  
Continuous Displacement

Continuous Displacement Cluster Variation Method is employed to study binary phase equilibria on the two dimensional square lattice with Lennard-Jones type pair potentials. It is confirmed that the transition temperature decreases significantly as compared with the one obtained by conventional Cluster Variation Method. This is ascribed to the distribution of atomic pairs in a wide range of atomic distance, which enables the system to attain the lower free energy. The spatial distribution of atomic species around a Bravais lattice point is visualized. Although the average position of an atom is centred at the Bravais lattice point, the maximum pair probability is not necessarily attained for the pairs located at the neighboring Bravais lattice points. In addition to the real space information, k-space information are calculated in the present study. Among them, the diffuse intensity spectra due to short range ordering and atomic displacement are discussed.

scholarly journals On the maximal circumradius of a planar convex set containing one lattice point

1995 ◽  
Vol 52 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Poh W. Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Set ◽  
Compact Convex ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

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