Two stationarity conditions in the $GI/r_0/M/r_1$ dam

M. Jankiewicz

Displaying similar documents to “Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam”

The scaling limits of a heavy tailed Markov renewal process

Julien Sohier (2013)

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

Similarity:

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.

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

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

Similarity:

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...

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

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

Similarity:

We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$ . The state space of $(Z, S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z, S)$ is the product of the uniform probability measure and a Gaussian distribution.

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

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...

On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$ -spaces

Jakub Olejnik (2010)

Banach Center Publications

Similarity:

In this paper we consider processes Xₜ with values in $L^{p}$ , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier, Michael Eckhoff, Véronique Gayrard, Markus Klein (2004)

Journal of the European Mathematical Society

Similarity:

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ 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 $ϵ ↓ 0$ , to the capacities of suitably constructed sets. We show that...

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

Similarity:

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...

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

Similarity:

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...

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Seiichiro Kusuoka, Carlo Marinelli (2014)

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

Similarity:

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $ℝ^{d}$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y = W \circ T$ , where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$ ) and of an arbitrary Lévy process independent of $Y$ , that the drift coefficient is continuous...

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

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

Similarity:

In the present work, we consider spectrally positive Lévy processes $(X_{t}, t \geq 0)$ not drifting to $+ \infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$ ) before hitting $0$ . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $ℙ_{x}^{☆}$ of this conditioned process (starting at $x g t; 0$ ) is defined as a Doob $h$ -transform via a martingale. For Lévy processes with infinite variation paths,...

Markov's property for kth derivative

Mirosław Baran, Beata Milówka, Paweł Ozorka (2012)

Annales Polonici Mathematici

Similarity:

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....

On the strong Brillinger-mixing property of $α$ -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

Similarity:

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C (x, y)$ defining an $α$ -determinantal point process (DPP). Assuming absolute integrability of the function $C_{0} (x) = C (o, x)$ , we show that a stationary $α$ -DPP with kernel function $C_{0} (x)$ is “strongly” Brillinger-mixing, implying, among others, that its tail- $σ$ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert, Florian Simatos (2014)

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

Similarity:

We consider regenerative processes with values in some general Polish space. We define their $ε$ -big excursions as excursions $e$ such that $ϕ (e) g t; ε$ , where $ϕ$ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$ . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $ε$ -big excursions and of their endpoints, for all $ε$ in a set whose closure contains $0$ . Finally,...

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

Similarity:

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.

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

Agnieszka Kowalska (2015)

Banach Center Publications

Similarity:

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...

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

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

Similarity:

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K (t), i (t), Y (t))$ on $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , where $𝕋^{2}$ is the two-dimensional torus. Here $(K (t), i (t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y (t)$ is an additive functional of $K$ , defined as $\int_{0}^{t} v (K (s)) d s$ , where $| v | \sim 1$ for small $k$ . We prove that the rescaled process ${(N ln N)}^{- 1 / 2} Y (N t)$ converges in distribution to a two-dimensional Brownian motion. As a consequence,...

Spectral condition, hitting times and Nash inequality

Eva Löcherbach, Oleg Loukianov, Dasha Loukianova (2014)

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

Similarity:

Let $X$ be a $μ$ -symmetric Hunt process on a LCCB space $𝙴$ . For an open set $𝙶 \subseteq 𝙴$ , let $τ_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$ . Let $r : [0, \infty [\to [0, \infty [$ and $R (t) = \int_{0}^{t} r (s) d s$ . We give necessary and sufficient conditions for $𝔼_{μ} R (τ_{𝙶}) l t; \infty$ in terms of the behavior near the origin of the spectral measure of $- A^{𝙶}$ . When $r (t) = t^{l}$ , $l \geq 0$ , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l + 1$ for $τ_{𝙶}$ ...

Limit distributions for multitype branching processes of $m$ -ary search trees

Brigitte Chauvin, Quansheng Liu, Nicolas Pouyanne (2014)

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

Similarity:

Let $m \geq 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m \leq 26$ , the asymptotic behavior of the process is Gaussian, but for $m \geq 27$ it is no longer Gaussian and a limit $W^{D T}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

Similarity:

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...

An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

Charles-Elie Rabier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval $[0, T]$ representing a chromosome. The originality is in the fact that some genotypes are missing. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on $[0, T]$ and under local alternatives with a QTL at $t^{☆}$ on $[0, T]$ . We show that the LRT process is asymptotically...

A remarkable $σ$ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano (2013)

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

Similarity:

The $σ$ -finite measure $𝒫_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$ -transform processes with respect to these functions are utilized for the construction of $𝒫_{sup}$ .

Displaying similar documents to “Two stationarity conditions in the G I / r 0 / M / r 1 dam”

Displaying similar documents to “Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam”