Recurrence for Markov processes on N lines

1971 ◽  
Vol 8 (4) ◽  
pp. 724-730
Author(s):  
Mark Pinsky
Keyword(s):  
Markov Process ◽  
Characteristic Function ◽  
Markov Processes ◽  
Real Line ◽  
Transition Function ◽  
Characteristic Functions ◽  
The Real ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Continuous Markov Process

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

Recurrence for Markov processes on N lines

1971 ◽  
Vol 8 (04) ◽  
pp. 724-730
Author(s):  
Mark Pinsky
Keyword(s):  
Markov Process ◽  
Characteristic Function ◽  
Markov Processes ◽  
Real Line ◽  
Transition Function ◽  
Characteristic Functions ◽  
The Real ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Continuous Markov Process

Let Λ = R 1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ (t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

scholarly journals An induced real quaternion spherical ensemble of random matrices

2017 ◽  
Vol 06 (01) ◽  
pp. 1750001
Author(s):  
Anthony Mays ◽  
Anita Ponsaing
Keyword(s):  
Real Line ◽  
Relative Size ◽  
Functional Differentiation ◽  
The Real ◽  
Real Quaternion ◽  
Large Matrix ◽  
Wall Boundary Conditions ◽  
The Matrix ◽  
Soft Wall ◽  
Point Correlation

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (01) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

An inequality for characteristic functions

1972 ◽  
Vol 6 (1) ◽  
pp. 1-9 ◽  
Author(s):  
C.R. Heathcote ◽  
J.W. Pitman
Keyword(s):  
Positive Integer ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
The Real ◽  
Powers Of 2

The paper is concerned with an extension of the inequality 1 - u(2nt) ≤ 4n[1-u(t)] for u(t) the real part of a characteristic function. The main result is that the inequality in fact holds for all positive integer n and not only powers of 2. Certain consequences are deduced and a brief discussion is given of the circumstances under which equality holds.

Creation of Mass Processes and Perturbation Theory

1973 ◽  
Vol 25 (3) ◽  
pp. 456-467
Author(s):  
Talma Leviatan
Keyword(s):  
Perturbation Theory ◽  
Markov Process ◽  
Markov Processes ◽  
Transition Function ◽  
Semigroup Of Operators ◽  
Starting Time ◽  
Terminal Time

Creation of mass processes were treated lately by several authors. The idea was to find some generalized Markov process that will correspond to a semigroup of operators which are not necessarily contraction operators (or equivalently to a quasi transition function which is not submarkov). It was G. A. Hunt [6] who first suggested the idea of Markov processes in which both the starting time and the terminal time are random. Such processes were constructed by Helms [4] and treated also by Nagasawa [12] and the author [10].

scholarly journals On the Effect of the Absorption Coefficient in a Differential Game of Pollution Control

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 961
Author(s):  
Ekaterina Marova ◽  
Ekaterina Gromova ◽  
Polina Barsuk ◽  
Anastasia Shagushina
Keyword(s):  
Characteristic Function ◽  
Absorption Coefficient ◽  
Differential Game ◽  
Pollution Control ◽  
Characteristic Functions ◽  
Ecological Situation ◽  
The Real ◽  
Irkutsk Region ◽  
Function Construction

We consider various approaches for a characteristic function construction on the example of an n players differential game of pollution control with a prescribed duration. We explore the effect of the presence of an absorption coefficient in the game on characteristic functions. As an illustration, we consider a game in which the parameters are calculated based on the real ecological situation of the Irkutsk region. For this game, we compute a number of characteristic functions and compare their properties.

scholarly journals Dunkl Translation and Uncentered Maximal Operator on the Real Line

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Chokri Abdelkefi ◽  
Mohamed Sifi
Keyword(s):  
Characteristic Function ◽  
Real Line ◽  
Maximal Operator ◽  
Weak Type ◽  
Dunkl Operator ◽  
The Real

We establish estimates of the Dunkl translation of the characteristic functionχ[−ɛ,ɛ],ɛ>0, and we prove that the uncentered maximal operator associated with the Dunkl operator is of weak type(1,1). As a consequence, we obtain theLp-boundedness of this operator for1<p≤+∞.

scholarly journals On convergence and compactness in variation with shift of discrete probability laws

2021 ◽  
Vol 8 (3) ◽  
pp. 385-393
Author(s):  
Ivan A. Alexeev ◽  
◽  
Alexey A. Khartov ◽  
Keyword(s):  
Real Line ◽  
Discrete Distribution ◽  
Distribution Functions ◽  
Divisible Distribution ◽  
Characteristic Functions ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
The Real ◽  
Convergence In Variation ◽  
Compactness Theorems

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

scholarly journals On Bounds for the Characteristic Functions of Some Degenerate Multidimensional Distributions

2003 ◽  
Vol 10 (2) ◽  
pp. 353-362
Author(s):  
T. Shervashidze
Keyword(s):  
Lower Bound ◽  
Characteristic Function ◽  
Chebyshev Polynomials ◽  
Random Variables ◽  
Characteristic Functions ◽  
Multidimensional Analogue ◽  
The Matrix ◽  
Trigonometric Integral

Abstract We discuss an application of an inequality for the modulus of the characteristic function of a system of monomials in random variables to the convergence of the density of the corresponding system of the sample mixed moments. Also, we consider the behavior of constants in the inequality for the characteristic function of a trigonometric analogue of the above-mentioned system when the random variables are independent and uniformly distributed. Both inequalities were derived earlier by the author from a multidimensional analogue of Vinogradov's inequality for a trigonometric integral. As a byproduct the lower bound for the spectrum of is obtained, where 𝐴𝑘 is the matrix of coefficients of the first 𝑘 + 1 Chebyshev polynomials of first kind.

Sign in / Sign up

Export Citation Format

Share Document