Gaussian approximation for functionals of Gibbs particle processes

Daniela Flimmel; Viktor Beneš

Displaying similar documents to “Gaussian approximation for functionals of Gibbs particle processes”

Stationary distributions for jump processes with memory

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

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

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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.

Estimating the conditional expectations for continuous time stationary processes

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

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One of the basic estimation problems for continuous time stationary processes $X_{t}$ , is that of estimating $E {X_{t + β} | X_{s} : s \in [0, t]}$ based on the observation of the single block ${X_{s} : s \in [0, t]}$ when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.

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

Jakub Olejnik (2010)

Banach Center Publications

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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...

Some properties of stationary sets

C. A. Di Prisco, W. Marek

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CONTENTSIntroduction..................................................................51. Derivative of a stationary set...................................72. Stationary degrees ...............................................133. Subsets of $P_{ϰ} (λ)$ ..............................................194. Stationary subsets of $P_{ϰ} (λ)$ .............................255. Superstationary substes of $P_{ϰ} (λ)$ ....................326. End-stationary subsets of $P_{ϰ} (λ)$ ......................34References................................................................37 ...

On reduction of two-parameter prediction problems

J. Friedrich, L. Klotz, M. Riedel (1995)

Studia Mathematica

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We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_{2}$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_{2} \subseteq G_{2}$ to results for weakly stationary processes on $G_{1} \times G_{2}$ with observed values on $G_{1} \times S_{2}$ . In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

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

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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 $𝒫_{sup}$ .

Sungularity as $L_{2}$ -regularity and vice versa

T. Pogany (1990)

Matematički Vesnik

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

Lothar Heinrich (2016)

Applications of Mathematics

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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...

On the divergence of certain integrals of the Wiener process

Lawrence A. Shepp, John R. Klauder, Hiroshi Ezawa (1974)

Annales de l'institut Fourier

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Let $f (x)$ be a nonnegative function with its only singularity at $x = 0$ , e.g. $f (x) = | x |^{- α}$ , $α > 0$ . We study the behavior of the Wiener process $W (t)$ in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on $f$ for the integrals of $f (W (t))$ to be finite or infinite.

Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

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

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Branching Processes in Random Environment (BPREs) $(Z_{n} : n \geq 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $ℙ (1 \leq Z_{n} \leq k | Z_{0} = i)$ , $k, i \in ℕ$ as $n \to \infty$ . More precisely, we characterize...

A note on the existence of Gibbs marked point processes with applications in stochastic geometry

Martina Petráková (2023)

Kybernetika

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This paper generalizes a recent existence result for infinite-volume marked Gibbs point processes. We try to use the existence theorem for two models from stochastic geometry. First, we show the existence of Gibbs facet processes in $ℝ^{d}$ with repulsive interactions. We also prove that the finite-volume Gibbs facet processes with attractive interactions need not exist. Afterwards, we study Gibbs-Laguerre tessellations of $ℝ^{2}$ . The mentioned existence result cannot be used, since one of its...

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

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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...

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

M. Jankiewicz (1988)

Applicationes Mathematicae

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Bernstein inequality for the parameter of the pth order autoregressive process AR(p)

Samir Benaissa (2006)

Applicationes Mathematicae

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The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X ̃_{t} = θ ̃ X ̃_{t - 1} + ε ̃_{t}$ . In this paper we study the convergence in distribution of the linear operator $I (θ ̃_{T}, θ ̃) = (θ ̃_{T} - θ ̃) θ ̃^{T - 2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator. ...