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...

Asymptotic properties of the sequences (a) ${P_{\phi }^{j}g}_{j=1}^{\infty }$ and (b) ${j^{-1}{\sum }_{i=0}^{j-1}P{ⁱ}_{\phi }g}_{j=1}^{\infty }$, where $P_{\phi }:L¹\to L¹$ 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...

In 1889 A. Markov proved that for every polynomial p in one variable the inequality $\left|\right|p^{\text{'}}{\left|\right|}_{[-1,1]}{\le \left(degp\right)²\left|\right|p\left|\right|}_{[-1,1]}$ 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...

Consider the normed space $(\mathbb{P}\left({ℂ}^N\right),|\left|·\right|\left|\right)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_g:(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|\left)\ni f\mapsto gf\in \right(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|\partial /\partial z_j{P\left|\right|\le M\left(degP\right)}^m\left|\right|P\left|\right|$, j = 1,...,N, $P\in \mathbb{P}\left({ℂ}^N\right)$, with positive constants M and m is equivalent to the inequality $\left|\right|{\partial }^N/\partial z₁...\partial z_{N}P\left|\right|\le M^{\text{'}}{\left(degP\right)}^{m^{\text{'}}}\left|\right|P\left|\right|$, $P\in \mathbb{P}\left({ℂ}^N\right)$, 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....

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...

Suppose that $\{Xn:n\ge 0\}$ 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 $Y_n:=N^{-1/2}{\sum }_{n=0}^{N}V\left(X_n\right)$ 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 $V\left(X_0\right)$ 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...

In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\alpha$-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.

This paper presents a novel approach for computing the strong Stackelberg/Nash equilibrium for Markov chains games. For solving the cooperative $n$-leaders and $m$-followers Markov game we consider the minimization of the $L_p-$norm that reduces the distance to the utopian point in the Euclidian space. Then, we reduce the optimization problem to find a Pareto optimal solution. We employ a bi-level programming method implemented by the extraproximal optimization approach for computing the strong...

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.

Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. 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 ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.

A notion of a wide-sense Markov process $X_t$ of order k ≥ 1, $X_t\sim WM\left(k\right)$, 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 $X_t\sim WM\left(k\right)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_t\sim WM\left(k\right)$ iff $x_t$ is a k-dimensional WM(1) process and iff the covariance function of $x_t$ has the triangular...

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 $\pi ,\tilde{\pi }$, some of the following charactristics are usually studied: $\parallel \pi -\tilde{\pi }{\parallel }_p$ for asymptotical stability [3], $|{\pi }_i-{\tilde{\pi }}_i|,\frac{|{\pi }_i-{\tilde{\pi }}_i|}{{\pi }_i}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P=P\left(t\right)$ and stationary probability vector $\pi \left(t\right)$, derivatives are also used...

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset {ℂ}^N$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function ${\Phi }_E$. Moreover, we show that one of these extremal-like functions is equal to ${\Phi }_E$ if E is a nonpluripolar set with $li{m}_{n\to \infty }Mₙ{\left(E\right)}^{1/n}=1$ where $Mₙ\left(E\right):=sup{\left|\right|\left|gradP\right|\left|\right|}_E/{\left|\right|P\left|\right|}_E$, the supremum is taken over all polynomials P of N variables...

For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M{ₙ}^{\alpha }={\left(A{ₙ}^{\alpha }\right)}^{-1}{\sum }_{k=0}^n{A}_{n-k}^{\alpha -1}{T}^k$. For α = 1, we find $Mₙ¹={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$, the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ 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....

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $-ϵ\Delta +\nabla F(·)\nabla$ on ${ℝ}^d$ or subsets of ${ℝ}^d$, where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ\downarrow 0$, to the capacities of suitably constructed sets. We show that...

In this paper we establish a decoupling feature of the random interlacement process ${ℐ}^u\subset {\mathbb{Z}}^d$ at level $u$, $d\ge 3$. Roughly speaking, we show that observations of ${ℐ}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of ${\mathbb{Z}}^d$ 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...

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${\left(Tⁿ\right)}_{n=1,2,...}$ by the continuous semigroup ${\left(e^{-t(I-T)}\right)}_{t\ge 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $supn\parallel Tⁿ-T^{n+1}\parallel :n=1,2,...<\infty$. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^p\left(\nu \right)$, $T₁,...,T_k$, $n̅=(n₁,...,n_k)\in {\mathbb{N}}^k$ and $\alpha ̅=(\alpha ₁,...,{\alpha }_k)$ with $0<{\alpha }_j\le 1$, we define the ergodic Cesàro-α̅ averages ${}_{n̅,\alpha ̅}f=1/\left({\prod }_{j=1}^k{A}_{n_j}^{{\alpha }_j}\right){\sum }_{i_k=0}^{n_k}\cdots {\sum }_{i₁=0}^{n₁}{\prod }_{j=1}^k{A}_{n_j-i_j}^{{\alpha }_j-1}{T}_k^{i_k}\cdots T{₁}^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^p\left(\nu \right)$ norm, when $n₁,...,n_k\to \infty$ independently, for all $f\in L^p\left(d\nu \right)$ with p > 1/α⁎ where $\alpha ⁎=mi{n}_{1\le j\le k}{\alpha }_j$. In the limit case p = 1/α⁎, we prove that the averages ${}_{n̅,\alpha ̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $\Lambda (1/\alpha ⁎,{\phi }_{m-1})$ with $\phi ₘ\left(t\right)=t{(1+log⁺t)}^m$. To obtain the result in the limit case we need...