A note on discriminating Poisson processes from other point processes with stationary inter arrival times

Gusztáv Morvai; Benjamin Weiss

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Displaying similar documents to “A note on discriminating Poisson processes from other point processes with stationary inter arrival times”

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...

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk (2011)

Colloquium Mathematicae

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Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality $x \in X : s u p_{n} | | \sum_{k = 1}^{n} T^{k} x | | < \infty$ = (I-T)X $.$ We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_{t} : t \geq 0$ with generator A satisfies $A X = x \in X : s u p_{s > 0} | | \int_{0}^{s} T_{t} x d t | | < \infty$ . The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities...

Prolongation of Poisson $2$ -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, $M$ denotes a smooth manifold of dimension $n$ , $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M, ω_{M})$ is a Poisson manifold with $2$ -form $ω_{M}$ , we construct the $2$ -Poisson form $ω_{M^{A}}^{A}$ , prolongation on $M^{A}$ of the $2$ -Poisson form $ω_{M}$ . We give a necessary and sufficient condition for that $M^{A}$ be an $A$ -Poisson manifold.

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.

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...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

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Let M be a smooth manifold of dimension m>0, and denote by $S_{c a n}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(T M, Π^{T}, S_{c a n})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r} M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A} M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

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.

Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

Dariusz Buraczewski, Ewa Damek (2010)

Colloquium Mathematicae

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Let N be a simply connected nilpotent Lie group and let $S = N ⋊ {(ℝ ⁺)}^{d}$ be a semidirect product, ${(ℝ ⁺)}^{d}$ acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that $l i m_{t \to \infty} t^{χ ₀} ν x : | x | > t = C > 0$ . In particular, this applies to classical Poisson kernels on symmetric...

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

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

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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,...

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...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof, Victor Ginzburg (2010)

Journal of the European Mathematical Society

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The hypersurface in $ℂ^{3}$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_{r}, r = 6, 7, 8$ , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_{r}$ provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra $ℂ [x_{1}, x_{2}, x_{3}]$ to a noncommutative algebra with generators $x_{1}, x_{2}, x_{3}$ and the following 3 relations...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

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Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k \in ℤ$ , if two functions vanish on ${SS}_{k} (F)$ , then so does their Poisson bracket.

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 $𝒫_{sup}$ .

Limit theorems for geometric functionals of Gibbs point processes

T. Schreiber, J. E. Yukich (2013)

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

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Observations are made on a point process $𝛯$ in $ℝ^{d}$ in a window $Q_{λ}$ of volume $λ$ . The observation, or ‘score’ at a point $x$ , here denoted $ξ (x, 𝛯)$ , is a function of the points within a random distance of $x$ . When the input $𝛯$ is a Poisson or binomial point process, the large $λ$ limit theory for the total score $\sum_{x \in 𝛯 \cap Q_{λ}} ξ (x, 𝛯 \cap Q_{λ})$ , when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian...

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

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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...

Universal rates for estimating the residual waiting time in an intermittent way

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

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A simple renewal process is a stochastic process ${X_{n}}$ taking values in ${0, 1}$ where the lengths of the runs of $1$ ’s between successive zeros are independent and identically distributed. After observing $X_{0}, X_{1}, ... X_{n}$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional...

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

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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,...