Scale vs. Translation Invariant Measures of Inequality of Opportunity when the Outcome is Binary

2013 ◽  
Author(s):  
Florian Wendelspiess Chávez Juárez ◽  
Isidro Soloaga
Keyword(s):  
Invariant Measures ◽  
Inequality Of Opportunity ◽  
Translation Invariant ◽  
Measures Of Inequality ◽  
Translation Invariant Measures

FUNCTIONS OF MARKOV PROCESSES AND ALGEBRAIC MEASURES

1992 ◽  
Vol 04 (01) ◽  
pp. 39-64 ◽  
Author(s):  
M. FANNES ◽  
B. NACHTERGAELE ◽  
L. SLEGERS
Keyword(s):  
Markov Processes ◽  
Invariant Measures ◽  
Canonical Representation ◽  
Positive Matrix ◽  
Translation Invariant ◽  
State Spaces ◽  
Finite State ◽  
The Mean ◽  
Class Of Functions ◽  
Translation Invariant Measures

We introduce a class of translation-invariant measures on the set {0, …, q−1}ℤ determined by a set of q d-dimensional matrices. They are algebraic in the sense that their densities are obtained by applying a functional to products of the defining matrices. Positivity of probabilities is assured by assuming a positivity structure on the algebra of defining matrices. Restricting attention to the usual positivity notion of positive matrix elements, a detailed analysis leads to a canonical representation theorem that solves the parametrization problem. Furthermore, we show that the class of algebraic measures coincides with the class of functions of Markov processes with finite state spaces. Our main result consists in the detailed study of the asymptotics of the conditional probabilities from which we derive a formula for the mean entropy.

Translation invariant measures which are not Hausdorff measures

1966 ◽  
Vol 62 (4) ◽  
pp. 693-698 ◽  
Author(s):  
K. E. Hirst
Keyword(s):  
Invariant Measures ◽  
Hausdorff Measures ◽  
Translation Invariant ◽  
Translation Invariant Measures

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.

scholarly journals On Comprehensively Intermediate Measures of Inequality and Poverty, with an Illustrative Application to Global Data

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
S. Subramanian
Keyword(s):  
Invariant Measures ◽  
Extreme Values ◽  
Scale Invariant ◽  
Relative Poverty ◽  
Poverty Measures ◽  
Variable Function ◽  
Measurement Of Inequality ◽  
Measures Of Inequality ◽  
Global Data ◽  
Illustrative Application

AbstractThe dominant convention in the measurement of inequality and poverty is to employ scale-invariant and replication-invariant measures, that is, measures that are thoroughgoingly relative. This is a routine feature of both the theoretical and applied literature in the area, despite weighty arguments that have been advanced by certain practitioners in favor of centrist measures which avoid the “extreme” values of both income-relative and income-absolute measures. The present paper extends these arguments in favor of measures which arebothincome-centrist and population centrist. A comprehensively centrist Gini coefficient of inequality is proposed, and likewise a comprehensively centrist class of poverty measures which are counterparts of the well-known Foster-Greer Thorbecke class of relative poverty measures. It is suggested that our diagnosis of the problems of inequality and poverty is likely to be a profoundly variable function of the precise types of inequality and poverty measures we employ in order to assess the magnitudes and trends of the phenomenon.

scholarly journals Invariant measures of interacting particle systems: Algebraic aspects

2020 ◽  
Vol 24 ◽  
pp. 526-580
Author(s):  
Luis Fredes ◽  
Jean-François Marckert
Keyword(s):  
Markov Process ◽  
Invariant Measures ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Invariant Distribution ◽  
Rate Matrix ◽  
Translation Invariant ◽  
Contact Processes ◽  
Jump Rate ◽  
Voter Models

Consider a continuous time particle system ηt = (ηt(k), k ∈ 𝕃), indexed by a lattice 𝕃 which will be either ℤ, ℤ∕nℤ, a segment {1, ⋯ , n}, or ℤd, and taking its values in the set Eκ𝕃 where Eκ = {0, ⋯ , κ − 1} for some fixed κ ∈{∞, 2, 3, ⋯ }. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix ⊤. These are standard settings, satisfied by the TASEP, the voter models, the contact processes. The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix ⊤ so that this Markov process admits some simple invariant distribution, as a product measure (if 𝕃 is any of the spaces mentioned above), the law of a Markov process indexed by ℤ or [1, n] ∩ ℤ (if 𝕃 = ℤ or {1, …, n}), or a Gibbs measure if 𝕃 = ℤ/nℤ. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory m). (As usual, a random process X indexed by ℤ or ℕ is said to be a Markov chain with memory m ∈ {0, 1, 2, ⋯ } if ℙ(Xk ∈ A | Xk−i, i ≥ 1) = ℙ(Xk ∈ A | Xk−i, 1 ≤ i ≤ m), for any k.) We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on ℤ2, with jump rates indexed by 2 × 2 squares, admitting product invariant measures.

scholarly journals Soliton Decomposition of the Box-Ball System

2021 ◽  
Vol 9 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Chi Nguyen ◽  
Leonardo T. Rolla ◽  
Minmin Wang
Keyword(s):  
Invariant Measures ◽  
Vries Equation ◽  
Linear Equations ◽  
Large Family ◽  
Ergodic Measure ◽  
System Of Linear Equations ◽  
Translation Invariant ◽  
Discrete Counterpart ◽  
Product Measures ◽  
Effective Speed

Abstract The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size k solitons in each k-slot. The dynamics of the components is linear: the kth component moves rigidly at speed k. Let $\zeta $ be a translation-invariant family of independent random vectors under a summability condition and $\eta $ be the ball configuration with components $\zeta $ . We show that the law of $\eta $ is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac {1}{2}$ . We also show that starting BBS with an ergodic measure, the position of a tagged k-soliton at time t, divided by t converges as $t\to \infty $ to an effective speed $v_k$ . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.

Sign in / Sign up

Export Citation Format

Share Document