scholarly journals Furstenberg entropy of intersectional invariant random subgroups

2018 ◽  
Vol 154 (10) ◽  
pp. 2239-2265
Author(s):  
Yair Hartman ◽  
Ariel Yadin
Keyword(s):  
Random Walk ◽  
General Framework ◽  
Discrete Group ◽  
A Priori ◽  
Probability Measures ◽  
Realization Problem ◽  
Schreier Graphs ◽  
Stationary Action ◽  
Lamplighter Groups ◽  
Invariant Random Subgroups

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

scholarly journals Subjective Values Theory: The Psychology of Value and Preferential Choice

2019 ◽  
Author(s):  
Dale Cohen ◽  
Amanda R. Cromley ◽  
Katelyn E. Freda ◽  
Madeline White
Keyword(s):  
Random Walk ◽  
Behavioral Economics ◽  
Prospect Theory ◽  
Utility Theory ◽  
Human Life ◽  
Sequential Sampling ◽  
A Priori ◽  
Decision Mechanism ◽  
Preferential Choice ◽  
Social Decisions

Here, we proposed Subjective Values Theory, a theory of the perception of value, andhow that perception drives preferential choice. Utility Theory, Prospect Theory, and traditional implementations of sequential sampling theory derive value from observers’ preferential choices. Subjective Values Theory goes beyond these theories by (a) precisely defining and measuring value independent of preferential choice, and (b) using these independent measurements of value to a priori predict preferential choice. We instantiate the decision mechanism proposed by Subjective Values Theory in a new Robust Random Walk (RRW) procedure. We evaluate the validity of Subjective Values Theory and the RRW in six experiments that measure the value of human lives and predict participants’ RTs and preferential choices in complex social decisions. In these experiments, we demonstrate that the process of perceiving Psychological Value is the same for objects and human lives, social status influences the perceived Psychological Value of a human life, and quantity has little or no influence on the perceived Psychological Value of human lives or objects. We discuss the implications of these findings in relation to decision theory, behavioral economics, and the psychology of morality.

Ch'ing Government and the Mineral Industries Before 1800

1968 ◽  
Vol 27 (4) ◽  
pp. 835-845 ◽  
Author(s):  
E-Tu Zen Sun
Keyword(s):  
Nineteenth Century ◽  
Government Policy ◽  
National Economy ◽  
General Framework ◽  
A Priori ◽  
Uniform Code ◽  
The Government ◽  
Mineral Industries

There were mines and smelters located in all the provinces of Ch'ing China before the nineteenth century. Their sizes varied greatly, as did their role in the local as well as national economy. Government policy regarding these industries was not based on a uniform code, but rather reflected a body of principles that served as guidelines for coping with each individual situation as it arose. Indeed, reading the early Ch'ing documents on mining leaves one at first with the impression that the authorities were eternally debating over the question, “to mine or not to mine.” Upon closer examination it soon becomes clear, however, that “mining affairs” were far from being administered along haphazard lines. It is the intention of this paper to describe some of the basic factors that went into the making of government decisions regarding mineral enterprises. In doing this, it would be helpful to keep these questions in mind: how did the mineral industries fit into the general framework of an agrarian-based civil bureaucracy? In the application of policy, did the government invariably let its actions be determined by a priori conceptions, or did it base decisions on pragmatic grounds? What sort of relation was generally accepted as the norm between the government and the operators of the enterprises in the instances where government interest was manifest?

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

scholarly journals An Optimized Cleaning Robot Path Generation and Execution System using Cellular Representation of Workspace

2020 ◽  
Author(s):  
Qile He ◽  
Yu Sun
Keyword(s):  
Random Walk ◽  
Path Planning ◽  
Work Environment ◽  
A Priori ◽  
Path Generation ◽  
Complete Coverage ◽  
Cellular Decomposition ◽  
Cleaning Robot ◽  
Robot Path

Many robot applications depend on solving the Complete Coverage Path Problem (CCPP). Specifically, robot vacuum cleaners have seen increased use in recent years, and some models offer room mapping capability using sensors such as LiDAR. With the addition of room mapping, applied robotic cleaning has begun to transition from random walk and heuristic path planning into an environment-aware approach. In this paper, a novel solution for pathfinding and navigation of indoor robot cleaners is proposed. The proposed solution plans a path from a priori cellular decomposition of the work environment. The planned path achieves complete coverage on the map and reduces duplicate coverage. The solution is implemented inside the ROS framework, and is validated with Gazebo simulation. Metrics to evaluate the performance of the proposed algorithm seek to evaluate the efficiency by speed, duplicate coverage and distance travelled.

scholarly journals Regularity results for solutions of linear elliptic degenerate boundary-value problems

2020 ◽  
Vol 9 (3) ◽  
pp. 545-566
Author(s):  
A. El Baraka ◽  
M. Masrour
Keyword(s):  
Sobolev Spaces ◽  
General Framework ◽  
A Priori ◽  
A Priori Estimate ◽  
Priori Estimate ◽  
Degenerate Elliptic ◽  
Type Spaces ◽  
Order Degenerate ◽  
Regularity Results ◽  
Besov Type Spaces

Abstract We give an a-priori estimate near the boundary for solutions of a class of higher order degenerate elliptic problems in the general Besov-type spaces $$B^{s,\tau }_{p,q}$$ B p , q s , τ . This paper extends the results found in Hölder spaces $$C^s$$ C s , Sobolev spaces $$H^s$$ H s and Besov spaces $$B^s_{p,q}$$ B p , q s , to the more general framework of Besov-type spaces.

scholarly journals Hausdorff spectrum of harmonic measure

2015 ◽  
Vol 37 (1) ◽  
pp. 277-307 ◽  
Author(s):  
RYOKICHI TANAKA
Keyword(s):  
Random Walk ◽  
Harmonic Measure ◽  
Hausdorff Measure ◽  
Hyperbolic Group ◽  
Volume Growth ◽  
Multifractal Spectrum ◽  
Parameter Family ◽  
Probability Measures ◽  
Dimensional Properties

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

scholarly journals A note on the geometric ergodicity of a Markov chain

1989 ◽  
Vol 21 (3) ◽  
pp. 702-704 ◽  
Author(s):  
K. S. Chan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Convergence Rates ◽  
Transition Probability ◽  
Invariant Probability Measure ◽  
Autoregressive Process ◽  
Probability Measures ◽  
Geometric Ergodicity ◽  
Drift Condition ◽  
Step Transition

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

scholarly journals Amenability of groupoids and asymptotic invariance of convolution powers

2021 ◽  
pp. 69-92
Author(s):  
Theo Bühler ◽  
Vadim Kaimanovich
Keyword(s):  
Random Walk ◽  
Locally Compact Group ◽  
Probability Measures ◽  
Locally Compact ◽  
Von Neumann ◽  
Original Definition ◽  
Convolution Powers ◽  
Measure Class ◽  
Definition Of

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

scholarly journals A note on American options with varying exercise price

1995 ◽  
Vol 37 (1) ◽  
pp. 45-57
Author(s):  
J. N. Dewynne ◽  
P. Wilmott
Keyword(s):  
Time Series ◽  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
A Priori ◽  
American Options ◽  
Share Price ◽  
Exercise Price ◽  
Black Scholes

AbstractWe examine the valuation of American options in a discrete time setting where the exercise price is known a priori but varies with time. (This is in contrast with the classical Black-Scholes [2] analysis, which lies in a continuous time framework and with constant exercise price.) In particular we consider a time series of exercise prices which are themselves a realisation of the share price random walk — that of the previous year, say.

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