scholarly journals Variable Length Memory Chains: Characterization of stationary probability measures

Bernoulli ◽  
2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Peggy Cénac ◽  
Brigitte Chauvin ◽  
Camille Noûs ◽  
Frédéric Paccaut ◽  
Nicolas Pouyanne
Keyword(s):  
Variable Length ◽  
Probability Measures ◽  
Stationary Probability

scholarly journals Random walks on projective spaces

2014 ◽  
Vol 150 (9) ◽  
pp. 1579-1606 ◽  
Author(s):  
Yves Benoist ◽  
Jean-François Quint
Keyword(s):  
Local Fields ◽  
Probability Measures ◽  
Stationary Probability ◽  
Semisimple Lie Group ◽  
Dimensional Representation ◽  
Dense Subgroup ◽  
Finite Dimensional Representation ◽  
Empirical Measures ◽  
Finite Dimensional ◽  
Strongly Irreducible

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group, $V$ be a finite-dimensional representation of $G$ and $\mu $ be a probability measure on $G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic $\mu $-stationary probability measures on the projective space $\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact $G$-orbits in $\mathbb{P}(V)$. When $V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic $\mu $-stationary measures on the flag variety of $G$.

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

Cardinal-Probabilistic Interaction Indices and their Applications: A Survey

2003 ◽  
Vol 7 (2) ◽  
pp. 79-85 ◽  
Author(s):  
Katsushige Fujimoto ◽  
Keyword(s):  
Choquet Integral ◽  
Axiomatic Characterization ◽  
Probability Measures ◽  
Choice Probability ◽  
Hierarchical Decomposition ◽  
Möbius Transform

The class of cardinal probabilistic interaction indices obtained as expected marginal interactions includes the Shapley, Banzhaf, and chaining interaction indices and the Möbius and co-Möbius transform so. We will survey cardinal-probabilistic interaction indices and their applications, focusing on axiomatic characterization of the class of cardinal-probabilistic interaction indices. We show that these typical cardinal-probabilistic interaction indices can be represented as the Stieltjes integrals with respect to choice-probability measures on [0,1]. We introduce a method for hierarchical decomposition of systems represented by the Choquet integral using interaction indices.

scholarly journals Weighted Regret-Based Likelihood: A New Approach to Describing Uncertainty

2015 ◽  
Vol 54 ◽  
pp. 471-492
Author(s):  
Joseph Y. Halpern
Keyword(s):  
Axiomatic Characterization ◽  
Lower Probability ◽  
Probability Measures ◽  
New Approach ◽  
Upper Probability ◽  
Weighted Probability ◽  
Natural Way

Recently, Halpern and Leung suggested representing uncertainty by a set of weighted probability measures, and suggested a way of making decisions based on this representation of uncertainty: maximizing weighted regret. Their paper does not answer an apparently simpler question: what it means, according to this representation of uncertainty, for an event E to be more likely than an event E'. In this paper, a notion of comparative likelihood when uncertainty is represented by a set of weighted probability measures is defined. It generalizes the ordering defined by probability (and by lower probability) in a natural way; a generalization of upper probability can also be defined. A complete axiomatic characterization of this notion of regret-based likelihood is given.

scholarly journals On actions of epimorphic subgroups on homogeneous spaces

2000 ◽  
Vol 20 (2) ◽  
pp. 567-592 ◽  
Author(s):  
NIMISH A. SHAH ◽  
BARAK WEISS
Keyword(s):  
Invariant Measure ◽  
Finite Volume ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Algebraic Characterization ◽  
Probability Measures ◽  
Invariant Subset ◽  
Orbit Closures

For an inclusion $F < G < L$ of connected real algebraic groups such that $F$ is epimorphic in $G$, we show that any closed $F$-invariant subset of $L/\Lambda$ is $G$-invariant, where $\Lambda$ is a lattice in $L$. This is a topological analogue of a result due to S. Mozes, that any finite $F$-invariant measure on $L/\Lambda$ is $G$-invariant.This result is established by proving the following result. If in addition $G$ is generated by unipotent elements, then there exists $a\in F$ such that the following holds. Let $U\subset F$ be the subgroup generated by all unipotent elements of $F$, $x\in L/\Lambda$, and $\lambda$ and $\mu$ denote the Haar probability measures on the homogeneous spaces $\overline{Ux}$ and $\overline{Gx}$, respectively (cf. Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as $n\to\infty$.We also give an algebraic characterization of algebraic subgroups $F<{\rm SL}_n(\mathbb{R})$ for which all orbit closures on ${\rm SL}_n(\mathbb{R})/{\rm SL}_n(\Z)$ are finite-volume almost homogeneous, namely the smallest observable subgroup of ${\rm SL}_n(\mathbb{R})$ containing $F$ should have no non-trivial algebraic characters defined over $\mathbb{R}$.

A CHARACTERIZATION OF PROBABILITY MEASURES IN TERMS OF WICK PRODUCT INEQUALITIES

2008 ◽  
Vol 11 (03) ◽  
pp. 377-391 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Variable ◽  
Wick Product ◽  
Probability Measures ◽  
Polynomial Functions ◽  
Finite Moments ◽  
Normally Distributed

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds: [Formula: see text] We show that this result can be extended to a random variable X, not necessary Gaussian, having an infinite support and finite moments of all orders, if and only if its Szegö–Jacobi sequence {ωk}k ≥ 1 is super-additive.

STATIONARY WEAK SOLUTIONS FOR STOCHASTIC 3D NAVIER–STOKES EQUATIONS WITH LÉVY NOISE

2012 ◽  
Vol 12 (01) ◽  
pp. 1150006 ◽  
Author(s):  
ZHAO DONG ◽  
WENBO V. LI ◽  
JIANLIANG ZHAI
Keyword(s):  
Weak Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Lévy Noise ◽  
Probability Measures ◽  
Stationary Probability ◽  
Navier Stokes Equations ◽  
Wiener Processes ◽  
The One ◽  
Levy Noise

We first study the existence of stationary weak solutions of stochastic 3D Navier–Stokes equations involving jumps, and the associated Galerkin stationary probability measures for this case. Then we present a comparison between the Galerkin stationary probability measures for the case driven by Lévy noise and the one driven by Wiener processes.

A characterization of probability measures in terms of semi-quantum operators

2019 ◽  
Vol 22 (02) ◽  
pp. 1950009 ◽  
Author(s):  
Gabriela Popa ◽  
Aurel I. Stan
Keyword(s):  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Random Variables ◽  
Finite Family ◽  
Probability Measures ◽  
Existence And Uniqueness Theorem ◽  
Quantum Operators ◽  
Definition Of ◽  
Finite Moments

A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.

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