Probability distribution solutions of a general linear equation of infinite order

Tomasz Kochanek; Janusz Morawiec

Displaying similar documents to “Probability distribution solutions of a general linear equation of infinite order”

The set of probability distribution solutions of a linear functional equation

Janusz Morawiec, Ludwig Reich (2008)

Annales Polonici Mathematici

Similarity:

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.

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

Tomasz Kochanek, Janusz Morawiec (2010)

Annales Polonici Mathematici

Similarity:

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.

On the powers of Voiculescu's circular element

Ferenc Oravecz (2001)

Studia Mathematica

Similarity:

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.

Corrigenda to “Operator semi-stable probability measures on $R^{N}$ ”

A. Luczak (1987)

Colloquium Mathematicae

Similarity:

Self-decomposable probability distributions on $R^{m}$

Kazimierz Urbanik (1969)

Applicationes Mathematicae

Similarity:

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

Similarity:

We deal with the linear functional equation (E) $g (x) = \sum_{i = 1}^{r} p_{i} g (c_{i} x)$ , where g:(0,∞) → (0,∞) is unknown, $(p ₁, . . ., p_{r})$ is a probability distribution, and $c_{i}$ ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

Small ball probability estimates in terms of width

Rafał Latała, Krzysztof Oleszkiewicz (2005)

Studia Mathematica

Similarity:

A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have $γ ₙ (s K) \leq {(2 s)}^{w ² / 4} γ ₙ (K)$ for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ , = φ: ℝ → ℝ|φ is continuous, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the...

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

Similarity:

A generalized form of the usual Lognormal distribution, denoted with $_{γ}$ , is introduced through the γ-order Normal distribution $_{γ}$ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of $_{γ}$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić (2014)

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

Similarity:

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...

Stochastic control optimal in the Kullback sense

Jan Šindelář, Igor Vajda, Miroslav Kárný (2008)

Kybernetika

Similarity:

The paper solves the problem of minimization of the Kullback divergence between a partially known and a completely known probability distribution. It considers two probability distributions of a random vector $(u_{1}, x_{1}, ..., u_{T}, x_{T})$ on a sample space of $2 T$ dimensions. One of the distributions is known, the other is known only partially. Namely, only the conditional probability distributions of $x_{τ}$ given $u_{1}, x_{1}, ..., u_{τ - 1}, x_{τ - 1}, u_{τ}$ are known for $τ = 1, ..., T$ . Our objective is to determine the remaining conditional probability distributions of $u_{τ}$ ...

Operator semistable probability measures on $R^{N}$

A. Łuczak (1981)

Colloquium Mathematicae

Similarity:

Limit theorems for random fields

Nguyen van Thu

Similarity:

CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

On two fragmentation schemes with algebraic splitting probability

M. Ghorbel, T. Huillet (2006)

Applicationes Mathematicae

Similarity:

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 Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let Ω be a countable infinite product $Ω ₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

The right tail exponent of the Tracy–Widom $β$ distribution

Laure Dumaz, Bálint Virág (2013)

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

Similarity:

The Tracy–Widom $β$ distribution is the large dimensional limit of the top eigenvalue of $β$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a \to \infty$ the tail of the Tracy–Widom distribution satisfies $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

On the limit distribution of the well-distribution measure of random binary sequences

Christoph Aistleitner (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We prove the existence of a limit distribution of the normalized well-distribution measure $W (E_{N}) / \sqrt{N}$ (as $N \to \infty$ ) for random binary sequences $E_{N}$ , by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

Premium evaluation for different loss distributions using utility theory

Harman Preet Singh Kapoor, Kanchan Jain (2011)

Discussiones Mathematicae Probability and Statistics

Similarity:

For any insurance contract to be mutually advantageous to the insurer and the insured, premium setting is an important task for an actuary. The maximum premium ( $P_{m a x})$ that an insured is willing to pay can be determined using utility theory. The main focus of this paper is to determine $P_{m a x}$ by considering different forms of the utility function. The loss random variable is assumed to follow different Statistical distributions viz Gamma, Beta, Exponential, Pareto, Weibull, Lognormal and Burr....

The Bayes choice of an experiment in estimating a success probability

Alicja Jokiel-Rokita, Ryszard Magiera (2002)

Applicationes Mathematicae

Similarity:

A Bayesian method of estimation of a success probability p is considered in the case when two experiments are available: individual Bernoulli (p) trials-the p-experiment-or products of r individual Bernoulli (p) trials-the $p^{r}$ -experiment. This problem has its roots in reliability, where one can test either single components or a system of r identical components. One of the problems considered is to find the degree r̃ of the $p^{r ̃}$ -experiment and the size m̃ of the p-experiment such that the...

Order relations in the set of probability distribution functions and their applications in queueing theory

Tomasz Rolski

Similarity:

CONTENTSIntroduction......................................................................................................................................... 51. n-Monotonic functions on (— ∞, ∞)........................................................................................... 62. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts...............................................................................................................