Upgrading Probability via Fractions of Events

Roman Frič; Martin Papčo

Displaying similar documents to “Upgrading Probability via Fractions of Events”

Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and $q$ -deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a $*$ -algebra equipped with a state) drawn from an ensemble of pair-wise...

On two fragmentation schemes with algebraic splitting probability

M. Ghorbel, T. Huillet (2006)

Applicationes Mathematicae

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Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass x₀ ∈ (0,1) undergoes splitting into b > 1 fragments of random sizes with some size-dependent probability p(x₀). With probability 1-p(x₀), this particle is left unchanged forever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable and unstable case with $p (x ₀) = x ₀^{a}$ and $p (x ₀) = 1 - x ₀^{a}$ respectively, for some a > 0. In the first (resp. second) case,...

On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

Henryk Gacki (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x (t, ω) = h (t, ω) + \int^{t + δ (t)} ₀ k (t, τ, ω) f (τ, x_{τ} (ω)) d τ$ , (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability...

Contributions to foundations of probability calculus on the basis of the modal logical calculus $M C^{ν}$ or $M C_{*}^{ν}$

A. Montanaro, A. Bressan (1983)

Rendiconti del Seminario Matematico della Università di Padova

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The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Probability distribution solutions of a general linear equation of infinite order, II

Tomasz Kochanek, Janusz Morawiec (2010)

Annales Polonici Mathematici

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Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $F (x) = \int_{Ω} F (τ (x, ω)) P (d ω)$ . We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.

Slowdown estimates and central limit theorem for random walks in random environment

Alain-Sol Sznitman (2000)

Journal of the European Mathematical Society

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This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on $ℤ^{d}$ , when $d > 2$ . We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important...

Further results on laws of large numbers for uncertain random variables

Feng Hu, Xiaoting Fu, Ziyi Qu, Zhaojun Zong (2023)

Kybernetika

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The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent...

Zero-one laws for graphs with edge probabilities decaying with distance. Part I

Saharon Shelah (2002)

Fundamenta Mathematicae

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Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{| i - j |} = 1 / {| i - j |}^{α}$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..

Excited against the tide: a random walk with competing drifts

Mark Holmes (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$ th cookie at a site determines the transition probabilities (to the left and right) for the $k$ th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that...

The set of probability distribution solutions of a linear functional equation

Janusz Morawiec, Ludwig Reich (2008)

Annales Polonici Mathematici

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Let (Ω,,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation $F (x) = \int_{Ω} F (τ (x, ω)) d P (ω)$ we determine the set of all its probability distribution solutions.

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson, Gennady Samorodnitsky (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ g t; 0$ that determines the fluctuations of the process. When $0 l t; κ l t; 2$ , the averaged distributions of the hitting times of the random walk converge to a $κ$ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true...

Zero-one laws for graphs with edge probabilities decaying with distance. Part II

Saharon Shelah (2005)

Fundamenta Mathematicae

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Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being $p_{| i - j |} = 1 / {| i - j |}^{α}$ , where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first...

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

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We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

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We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $(T^{'})$ . We show that for every $ϵ > 0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp (- {(log n)}^{d - ϵ})$ . This bound almost matches the known lower bound of $exp (- C {(log n)}^{d})$ , and signiﬁcantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched...

Zeros of random functions in Bergman spaces

Joel H. Shapiro (1979)

Annales de l'institut Fourier

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Suppose $μ$ is a finite positive rotation invariant Borel measure on the open unit disc $Δ$ , and that the unit circle lies in the closed support of $μ$ . For $0 < p < \infty$ the $A_{μ}^{p}$ is the collection of functions in $L^{p} (μ)$ holomorphic on $Δ$ . We show that whenever a Gaussian power series $f (z) = Σ ζ_{n} a_{n} z^{n}$ almost surely lies in $A_{μ}^{p}$ but not in $⋃_{q > p} A_{μ}^{p}$ , then almost surely: a) the zero set $Z (f)$ of $f$ is not contained in any $A_{μ}^{q}$ zero set ( $q > p$ , and b) $Z (f + 1) \cup Z (f - 1)$ is not contained in any $A_{μ}^{q}$ zero set.

On the powers of Voiculescu's circular element

Ferenc Oravecz (2001)

Studia Mathematica

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The main result of the paper is that for a circular element c in a C*-probability space, $(c ⁿ, c^{n *})$ is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.

On the paradox of three random variables

S. Trybuła (1961)

Applicationes Mathematicae

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On the paradox of n random variables

S. Trybuła (1965)

Applicationes Mathematicae

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