Eigenvalues of operators in $L_p$-spaces in Markov chains with a general state space

Zbyněk Šidák

Displaying similar documents to “Eigenvalues of operators in $L_{p}$ -spaces in Markov chains with a general state space”

Evaluating default priors with a generalization of Eaton’s Markov chain

Brian P. Shea, Galin L. Jones (2014)

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

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We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$ . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

Tangential Markov inequality in $L^{p}$ norms

Agnieszka Kowalska (2015)

Banach Center Publications

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In 1889 A. Markov proved that for every polynomial p in one variable the inequality $| | p^{'} {| |}_{[- 1, 1]} {\leq (d e g p) ² | | p | |}_{[- 1, 1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs...

Insensitivity analysis of Markov chains

Kocurek, Martin

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Sensitivity analysis of irreducible Markov chains considers an original Markov chain with transition probability matix $P$ and modified Markov chain with transition probability matrix $P$ . For their respective stationary probability vectors $π, \tilde{π}$ , some of the following charactristics are usually studied: $∥ π - \tilde{π} ∥_{p}$ for asymptotical stability [3], $| π_{i} - {\tilde{π}}_{i} |, \frac{| π_{i} - {\tilde{π}}_{i} |}{π_{i}}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P = P (t)$ and stationary probability vector $π (t)$ , derivatives are also used...

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Suppose that ${X n : n \geq 0}$ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_{n} : = N^{- 1 / 2} \sum_{n = 0}^{N} V (X_{n})$ converge in law to a normal random variable, as $N \to + \infty$ . For a stationary Markov chain with the $L^{2}$ spectral gap the theorem holds for all $V$ such that $V (X_{0})$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds...

Markov's property for kth derivative

Mirosław Baran, Beata Milówka, Paweł Ozorka (2012)

Annales Polonici Mathematici

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Consider the normed space $(ℙ (ℂ^{N}), | | \cdot | |)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_{g} : (ℙ (ℂ^{N}), | | \cdot | |) ∋ f \mapsto g f \in (ℙ (ℂ^{N}), | | \cdot | |)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $| \partial / \partial z_{j} {P | | \leq M (d e g P)}^{m} | | P | |$ , j = 1,...,N, $P \in ℙ (ℂ^{N})$ , with positive constants M and m is equivalent to the inequality $| | \partial^{N} / \partial z ₁ . . . \partial z_{N} P | | \leq M^{'} {(d e g P)}^{m^{'}} | | P | |$ , $P \in ℙ (ℂ^{N})$ , with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras....

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Loïc Hervé, Françoise Pène (2010)

Bulletin de la Société Mathématique de France

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The Nagaev-Guivarc’h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order...

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

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Let E be a compact set in the complex plane, $g_{E}$ be the Green function of the unbounded component of $ℂ_{\infty} ∖ E$ with pole at infinity and $M ₙ (E) = s u p (| | P^{'} {| |}_{E} {) / (| | P | |}_{E})$ where the supremum is taken over all polynomials ${P |}_{E} ≢ 0$ of degree at most n, and ${| | f | |}_{E} = s u p | f (z) | : z \in E$ . The paper deals with recent results concerning a connection between the smoothness of $g_{E}$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${M ₙ (E)}_{n = 1, 2, . . .}$ . Some additional conditions are given for special classes of sets.

Compact subsets of $C^{n}$ preserving Markov’s inequality

W. Plesniak (1988)

Matematički Vesnik

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Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process $X_{t}$ of order k ≥ 1, $X_{t} \sim W M (k)$ , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_{t}$ is the k-dimensional process $x_{t} = (X_{t - k + 1}, . . ., X_{t})$ . The covariance structure of $X_{t} \sim W M (k)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_{t} \sim W M (k)$ iff $x_{t}$ is a k-dimensional WM(1) process and iff the covariance function of $x_{t}$ has the triangular...

Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam

M. Jankiewicz (1988)

Applicationes Mathematicae

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The scaling limits of a heavy tailed Markov renewal process

Julien Sohier (2013)

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

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In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $α$ -stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0, \infty) \times [0, a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process $ℐ^{u} \subset ℤ^{d}$ at level $u$ , $d \geq 3$ . Roughly speaking, we show that observations of $ℐ^{u}$ restricted to two disjoint subsets $A_{1}$ and $A_{2}$ of $ℤ^{d}$ are approximately independent, once we add a sprinkling to the process $ℐ^{u}$ by slightly increasing the parameter $u$ . Our results differ from previous ones in that we allow the mutual distance between the sets $A_{1}$ and $A_{2}$ to be much smaller than their diameters. We then provide an important application...

Acknowledgment to the paper: “Another proof that a chain of non-empty $H$ -closed subspaces has a non-empty intersection’

J. Mioduszewski (1971)

Colloquium Mathematicae

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Mean lower bounds for Markov operators

Eduard Emel&#039;yanov, Manfred Wolff (2004)

Annales Polonici Mathematici

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Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $(n^{- 1} \sum_{k = 0}^{n - 1} T^{k}) ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $l i m_{n \to \infty} | | (h - n^{- 1} \sum_{k = 0}^{n - 1} T^{k} f) ₊ | | = 0$ for every density f. Analogous results for strongly continuous semigroups are given.

L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

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Let Ω be an open subset of $ℝ^{d}$ with 0 ∈ Ω. Furthermore, let $H_{Ω} = - \sum_{i, j = 1}^{d} \partial_{i} c_{i j} \partial_{j}$ be a second-order partial differential operator with domain $C_{c}^{\infty} (Ω)$ where the coefficients $c_{i j} \in W_{l o c}^{1, \infty} (Ω ̅)$ are real, $c_{i j} = c_{j i}$ and the coefficient matrix $C = (c_{i j})$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If $\int_{0}^{\infty} d s s^{d / 2} e^{- λ μ (s) ²} < \infty$ for some λ > 0 where $μ (s) = \int_{0}^{s} d t c {(t)}^{- 1 / 2}$ then we establish that $H_{Ω}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....

Invariant densities for random $β$ -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ (β - 1)]$ . We show existence and uniqueness of a $K_{β}$ -invariant probability measure, absolutely continuous with respect to $m_{p} \otimes λ$ , where $m_{p}$ is the Bernoulli measure on ${0, 1}^{ℕ}$ with parameter $p$ ( $0 < p < 1$ ) and $λ$ is the normalized Lebesgue measure on $[0, ⌊ β ⌋ (β - 1)]$ . Furthermore, this measure is of the form $m_{p} \otimes μ_{β, p}$ , where $μ_{β, p}$ is equivalent to $λ$ . We prove that the measure of maximal entropy and $m_{p} \otimes λ$ are mutually...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

Sets with the Bernstein and generalized Markov properties

Mirosław Baran, Agnieszka Kowalska (2014)

Annales Polonici Mathematici

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It is known that for $C^{\infty}$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty}$ determining. In this paper we give examples of sets which are not $C^{\infty}$ determining, but have the Bernstein and generalized Markov properties.

Displaying similar documents to “Eigenvalues of operators in L p -spaces in Markov chains with a general state space”

Displaying similar documents to “Eigenvalues of operators in $L_{p}$ -spaces in Markov chains with a general state space”