391 results on '"60B12"'
Search Results
102. Extremes of weighted Brownian Bridges in increasing dimension.
- Author
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Jirak, Moritz
- Subjects
WIENER processes ,ASYMPTOTIC distribution ,EXTREME value theory ,EXTREMAL combinatorics ,EXTREMAL problems (Mathematics) ,APPROXIMATION algorithms ,WEIGHTED graphs - Abstract
Let $\bigl\{B_j(t)\bigr\}_{1 \leq j \leq N}$ be a sequence of independent Brownian Bridges and consider the quantity $\mathcal{V}_{n,N} = \sup_{l(n) \leq t \leq 1 - r(n)} \sqrt{\sum_{j = 1}^N B_j^2(t)}/\sqrt{t(1 - t)},$ where lim r( n), l( n) = 0. The random variable $\mathcal{V}_{n,N}$ plays a crucial role in many statistical applications and has been studied intensively over the past decades. As it turned out, $\mathcal{V}_{n,N}$ exhibits Darling-Erdös type behavior, i.e. for appropriate sequences a( n), b( n, N) we have $a(n)\mathcal{V}_{n,N} - b(n,N) \xrightarrow{w} \mathcal{G}$ as n→ ∞ and N remains fixed, where $\mathcal{G}$ is an extreme value distribution. We generalize this result by showing that an analogous version is still valid if $n = n(N) = \mathcal{O}\bigl(N^{q}\bigr)$ for arbitrary q > 0, where N→ ∞. The proof is based on a strong approximation result for squared Ornstein-Uhlenbeck processes which has some own interest. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
103. A law of large numbers approximation for Markov population processes with countably many types.
- Author
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Barbour, A. and Luczak, M.
- Subjects
- *
DYNAMICS , *APPROXIMATION theory , *MARKOV processes , *PARASITIC diseases , *PROOF theory , *STOCHASTIC convergence , *MATHEMATICAL models - Abstract
When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for quite general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted ℓ norm. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
104. Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces.
- Author
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Rosalsky, Andrew and Thanh, Le
- Subjects
- *
LAW of large numbers , *MARTINGALES (Mathematics) , *STOCHASTIC processes , *BANACH spaces , *MATHEMATICAL sequences , *STOCHASTIC convergence - Abstract
For a blockwise martingale difference sequence of random elements { V, n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form $$\lim _{n \to \infty } \sum\nolimits_{i = 1}^n {V_i /g_n = 0}$$ almost surely to hold where the constants g ↑ ∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 < p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
105. Complete convergence for weighted sums of arrays of banach-space-valued random elements*.
- Author
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Qiu, De, Hub, Tien-Chung, Cabrera, Manuel, and Volodin, Andrei
- Subjects
- *
BANACH spaces , *STOCHASTIC convergence , *VECTOR spaces , *TOPOLOGY , *MATHEMATICAL functions - Abstract
We study the complete convergence for weighted sums of arrays of Banach-space-valued random elements and obtain some new results that extend and improve the related known works in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
106. A Universality Property of Gaussian Analytic Functions.
- Author
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Ledoan, Andrew, Merkli, Marco, and Starr, Shannon
- Abstract
We consider random analytic functions defined on the unit disk of the complex plane $f(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}$, where the X's are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a are chosen so that f( z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and $\mathbf{E}f(z)\overline{f(w)}$ is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
107. An addendum to a remark on Slutsky's theorem
- Author
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Delbaen, Freddy, Azéma, Jacques, editor, Émery, Michel, editor, Ledoux, Michel, editor, and Yor, Marc, editor
- Published
- 1999
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108. A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers.
- Author
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Li, Deli, Qi, Yongcheng, and Rosalsky, Andrew
- Abstract
Let { X; n≥1} be a sequence of independent copies of a real-valued random variable X and set S= X+⋅⋅⋅+ X, n≥1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0< p<2, if and only if where $u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}$, n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159-186, ) inequality to obtain some general results for sums of the form $\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|$ (where { V; n≥1} is a sequence of independent Banach-space-valued random variables, and a≥0, n≥1), which may be of independent interest, but which we apply to $\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})$. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
109. Weak Convergence Results for Multiple Generations of a Branching Process.
- Author
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Kuelbs, James and Vidyashankar, Anand
- Abstract
We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling-Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space ( C[0,1]), with the product topology, or in Banach subspaces of ( C[0,1]) determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling-Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
110. Rates of convergence in the CLT for linear random fields.
- Author
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Mielkaitis, Edgaras and Paulauskas, Vygantas
- Subjects
- *
RANDOM fields , *HILBERT space , *APPROXIMATION theory , *GAUSSIAN processes , *LOGARITHMS - Abstract
In this paper, we study sums of linear random fields defined on the lattice Z with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091-1098, (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry-Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171-174, ] for linear processes. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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111. Pruitt's Estimates in Banach Space.
- Author
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Griffin, Philip
- Abstract
Pruitt's estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite-dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
112. Large deviation for stochastic line integrals as L p-currents.
- Author
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Kusuoka, Shigeo, Kuwada, Kazumasa, and Tamura, Yozo
- Subjects
- *
LARGE deviations (Mathematics) , *STOCHASTIC processes , *RIEMANNIAN manifolds , *SOBOLEV spaces , *WIENER processes - Abstract
The large deviation principle for stochastic line integrals along Brownian paths on a compact Riemannian manifold is studied. We regard them as a random map on a Sobolev space of 1-forms. We show that the differentiability order of the Sobolev space can be chosen to be almost independent of the dimension of the underlying space by assigning higher integrability on 1-forms. The large deviation is formulated for the joint distribution of stochastic line integrals and the empirical distribution of a Brownian path. As the result, the rate function is given explicitly. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
113. The Marcinkiewicz–Zygmund LLN in Banach Spaces: A Generalized Martingale Approach.
- Author
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Hechner, Florian and Heinkel, Bernard
- Abstract
A result due to Gut asserts that the Marcinkiewicz–Zygmund strong law of large numbers for real-valued random variables is an amart a.s. convergence property. In this paper, a necessary and sufficient condition is given, under which that SLLN is also a quasimartingale. We also study the case of Banach-space valued r.v. and show that the scalar result remains true when the space is of suitable stable type. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
114. Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces.
- Author
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Dung, Le, Ngamkham, Thuntida, Tien, Nguyen, and Volodin, A.
- Abstract
In this paper we establish Marcinkiewicz-Zygmund type laws of large numbers for double arrays of random elements in Banach spaces. Our results extend those of Hong and Volodin [6]. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
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115. Departure from normality of increasing-dimension martingales
- Author
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Arbués, Ignacio
- Subjects
- *
CENTRAL limit theorem , *LIMIT theorems , *ASYMPTOTIC distribution , *AUTOCORRELATION (Statistics) - Abstract
Abstract: In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the -dimensional average of martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR() and the order of the model grows with the length of the series. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
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116. A Gaussian Process Arising from Likert-Type Scaling
- Author
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Bhattacharya, Tuhinsubhra and Sengupta, Arindam
- Published
- 2017
- Full Text
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117. A Method for Approximating the Variance of the Sojourn Times in Star-Shaped Queueing Networks.
- Author
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Mei, R. D.van der, de Wilde, A. R., and Bhulai, S.
- Subjects
- *
QUEUEING networks , *CLIENT/SERVER computing , *DATA transmission systems simulations , *QUEUING theory , *COMPUTER network architectures , *PEER-to-peer architecture (Computer networks) - Abstract
We study the sojourn times in an open star-shaped queueing network, with a central processor-sharing (PS) node and multiple multi-server First-Come-First Served (FCFS) nodes. Each customer alternatingly visits the central node and one of the other nodes, before departing from the system. For this model, exact expressions for the mean sojourn time can be easily obtained, but an exact analysis of the variance is not possible. Therefore, we propose a method for deriving simple but accurate approximations for the variance of the sojourn times. Extensive simulations demonstrate that the approximations are extremely accurate for a wide range of parameter values. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
118. On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements.
- Author
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Kim, Tae-Sung and Ko, Mi-Hwa
- Abstract
Let { Y
i ;−∞< i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let { ai ;−∞< i<∞} be an absolutely summable sequence of real numbers and set Vi =∑ ai+ k Yi , i≥1. In this paper, we derive that if $n^{-\frac{1}{\mu}}\sum_{i=1}^{n}V_{i}\rightarrow^{p}0$ and E| X|μ logρ | X|<0, for some μ (0< μ<2, μ≠1) and ρ>0 then $\sum_{n=1}^{\infty}n^{-1}P\{\|\sum_{i=1}^{n}V_{i}\|>\epsilon n^{\frac{1}{\mu}}\}<\infty$ for all ε>0. [ABSTRACT FROM AUTHOR]- Published
- 2008
- Full Text
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119. Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion.
- Author
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Jolis, Maria and Viles, Noèlia
- Abstract
We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of $$B^{H_0}$$ , when H tends to H
0 . [ABSTRACT FROM AUTHOR]- Published
- 2007
- Full Text
- View/download PDF
120. Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation
- Author
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Davit Martirosyan, Vahagn Nersesyan, Laboratoire de Mathématiques de Versailles (LMV), and Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Nonlinear wave equation ,White in time noise ,35L70 ,01 natural sciences ,Upper and lower bounds ,010104 statistics & probability ,symbols.namesake ,Coupling method ,35R60 ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Nonlinear system ,Large deviations principle ,Fourier transform ,Regularization (physics) ,Dissipative system ,symbols ,Large deviations theory ,Preprint ,Statistics, Probability and Uncertainty ,Rate function ,60B12 ,60F10 - Abstract
International audience; We consider the damped nonlinear wave (NLW) equation driven by a noise which is white in time and colored in space. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and is a novelty in the context of randomly forced PDE’s. The proof is based on an extension of methods developed in (Comm. Pure Appl. Math. 68 (12) (2015) 2108–2143) and (Large deviations and mixing for dissipative PDE’s with unbounded random kicks (2014) Preprint) in the case of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system and the complex Ginzburg–Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere).
- Published
- 2018
121. Escape from the boundary in Markov population processes
- Author
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Andrew Barbour, Fima C. Klebaner, Haya Kaspi, Kais Hamza, and University of Zurich
- Subjects
Statistics and Probability ,Differential equation ,Population ,Markov process ,Boundary (topology) ,Scale (descriptive set theory) ,01 natural sciences ,boundary behaviour ,010104 statistics & probability ,symbols.namesake ,510 Mathematics ,60J27 ,2604 Applied Mathematics ,92D30 ,FOS: Mathematics ,Statistical physics ,2613 Statistics and Probability ,0101 mathematics ,education ,92D30 (Primary), 60J27, 60B12 (Secondary) ,Mathematics ,Branching process ,education.field_of_study ,Markov chain ,Markov population process ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,branching process ,10123 Institute of Mathematics ,Diffusion process ,symbols ,60B12 ,Mathematics - Probability - Abstract
Density dependent Markov population processes in large populations of size $N$ were shown by Kurtz (1970, 1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, and to exhibit stochastic fluctuations about this deterministic solution on the scale $\sqrt N$ that can be approximated by a diffusion process. Here, motivated by an example from evolutionary biology, we are concerned with describing how such a process leaves an absorbing boundary. Initially, one or more of the populations is of size much smaller than $N$, and the length of time taken until all populations have sizes comparable to $N$ then becomes infinite as $N \to \infty$. Under suitable assumptions, we show that in the early stages of development, up to the time when all populations have sizes at least $N^{1-\alpha}$, for $1/3 < \alpha < 1$, the process can be accurately approximated in total variation by a Markov branching process. Thereafter, the process is well approximated by the deterministic solution starting from the original initial point, but with a random time delay. Analogous behaviour is also established for a Markov process approaching an equilibrium on a boundary, where one or more of the populations become extinct., Comment: 50 pages
- Published
- 2015
122. Dominating points and large deviations for random vectors.
- Author
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Einmahl, U. and Kuelbs, J.
- Abstract
We establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a Banach space. These estimates are based on the existence of dominating points in this setting. [ABSTRACT FROM AUTHOR]
- Published
- 1996
- Full Text
- View/download PDF
123. Some remarks on a question of Strassen.
- Author
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Griffin, P. and Kuelbs, J.
- Abstract
Strassen's original functional law of the iterated logarithm for partial sums and Brownian motion examined convergence and clustering in the sup-norm. Here we address what happens if we use the much larger H-norm. We provide the answer to a query which appeared at the end of Strassen's original paper, and also present several contrasting results which are shown to be essentially best possible. [ABSTRACT FROM AUTHOR]
- Published
- 1996
- Full Text
- View/download PDF
124. The Gaussian measure of shifted balls.
- Author
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Kuelbs, James, Li, Wenbo, and Linde, Werner
- Abstract
Let μ be a centered Gaussian measure on a Hilbert space H and let $$B_R \subseteq H$$ be the centered ball of radius R>0. For a∈ H and $$\mathop {\lim }\limits_{t{\mathbf{ }} \to {\mathbf{ }}\infty } {\mathbf{ }}R(t)/t< {\mathbf{ }}||a||$$ , we give the exact asymptotics of μ( B+ t· a) as t→∞. Also, upper and lower bounds are given when μ is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates. [ABSTRACT FROM AUTHOR]
- Published
- 1994
- Full Text
- View/download PDF
125. Critical fluctuations of sums of weakly dependent random vectors.
- Author
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Wang, Kongming
- Abstract
Let S be sums of iid random vectors taking values in a Banach space and F be a smooth function. We study the fluctuations of S under the transformed measure P given by d P/ d P=exp ( nF( S/ n))/ Z. If degeneracy occurs then the projection of S onto the degenerate subspace, properly centered and scaled, converges to a non-Gaussian probability measure with the degenerate subspace as its support. The projection of S onto the non-degenerate subspace, scaled with the usual order $$\sqrt {n,} $$ converges to a Gaussian probability measure with the non-degenerate subspace as its support. The two projective limits are in general dependent. We apply this theory to the critical mean field Heisenberg model and prove a central limit type theorem for the empirical measure of this model. [ABSTRACT FROM AUTHOR]
- Published
- 1994
- Full Text
- View/download PDF
126. On smoothness conditions and convergence rates in the CLT in Banach spaces.
- Author
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Bentkus, Vidmantas and Götze, Friedrich
- Abstract
In Banach spaces the rate of convergence in the Central Limit Theorem is of order O(n) for sets which have 'regular' boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals. [ABSTRACT FROM AUTHOR]
- Published
- 1993
- Full Text
- View/download PDF
127. Stochastic approximation of quasi-stationary distributions on compact spaces and applications
- Author
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Bertrand Cloez, Michel Benaïm, Fabien Panloup, Institut de Mathématiques (UNINE), Université de Neuchâtel (UNINE), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), PANORisk, Institut de Mathematiques, and SNF : 200020/149871, 200021/175728
- Subjects
Statistics and Probability ,reinforced random walks ,random perturba- tions of dynamical systems ,Euler scheme ,Quasi-stationary distributions ,Boundary (topology) ,Markov process ,Stochastic approximation ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,symbols.namesake ,Position (vector) ,stochastic approximation ,spectral gap ,Secondary 34F05 ,FOS: Mathematics ,random perturba-tions of dynamical systems ,Applied mathematics ,0101 mathematics ,60J20 ,Mathematics ,60J60 ,Euler scheme AMS-MSC 65C20 ,Markov chain ,010102 general mathematics ,Probability (math.PR) ,random perturbations of dynamical systems ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Compact space ,34F05 ,symbols ,60J10 ,Spectral gap ,65C20 ,Statistics, Probability and Uncertainty ,extinction rate ,60B12 ,Mathematics - Probability - Abstract
International audience; In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.
- Published
- 2018
128. Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices
- Author
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Mireille Capitaine, Catherine Donati-Martin, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Pure mathematics ,Nonuniversality ,Gaussian ,eigenvectors ,Poincaré inequality ,15A18, 60F05 ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Matrix (mathematics) ,60F05 ,FOS: Mathematics ,Outliers ,Fluctuations ,0101 mathematics ,Eigenvalues and eigenvectors ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,46L54 ,010102 general mathematics ,Mathematics::Rings and Algebras ,Probability (math.PR) ,15A52 ,Block matrix ,16. Peace & justice ,Hermitian matrix ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Outlier ,symbols ,Random matrices ,Free probability. Mathematics Subject Classification 2000: 15A18 ,60B12 ,Random matrix ,Mathematics - Probability - Abstract
In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked $N\times N$ complex Deformed Wigner matrix $M_N$: $M_N =W_N/\sqrt{N} + A_N$ where $W_N$ is an $N \times N$ Hermitian Wigner matrix whose entries have a law $\mu$ satisfying a Poincar\'e inequality and the matrix $A_N$ is a block diagonal matrix, with an eigenvalue $\theta$ of multiplicity one, generating an outlier in the spectrum of $M_N$. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of $M_N$ onto a unit eigenvector corresponding to $\theta$ are not universal., Comment: accepted for publication in Latin American Journal of Probability and Mathematical Statistics
- Published
- 2018
- Full Text
- View/download PDF
129. A new approach for the construction of a Wasserstein diffusion
- Author
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Victor Marx, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,differential calculus on Wasserstein space ,Pure mathematics ,82B21 ,Space (mathematics) ,01 natural sciences ,010104 statistics & probability ,FOS: Mathematics ,Limit (mathematics) ,60G44 ,0101 mathematics ,Itô formula for measure-valued processes ,Brownian motion ,Mathematics ,Probability measure ,60J60 ,Sequence ,coalescing particles ,modified Arratia flow ,Interacting particle system ,Probability (math.PR) ,010102 general mathematics ,measure-valued processes ,Brownian sheet ,MSC 2010: Primary 60K35, 60J60, 60B12, Secondary 60G44, 82B21 ,Moment (mathematics) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Diffusion process ,60K35 ,Wasserstein diffusion ,Statistics, Probability and Uncertainty ,60B12 ,Mathematics - Probability ,interacting particle system - Abstract
International audience; We propose in this paper a construction of a diffusion process on the Wasserstein space P_2(R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. "A system of coalescing heavy diffusion particles on the real line", 2017) and consists of the limit when N tends to infinity of a system of N coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on P_2(R) constructed by von Renesse and Sturm (see Entropic measure and Wasserstein diffusion). We obtain that process by the construction of a system of particles having short-range interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.
- Published
- 2018
130. Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications
- Author
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Axel Munk, Max Sommerfeld, and Carla Tameling
- Subjects
Statistics and Probability ,limit law ,90C08 ,Scale (descriptive set theory) ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Space (mathematics) ,01 natural sciences ,empirical process ,010104 statistics & probability ,statistical testing ,60F05 ,FOS: Mathematics ,Optimal transport ,Countable set ,Applied mathematics ,Wasserstein distance ,0101 mathematics ,Empirical process ,Probability measure ,Mathematics ,62E20 ,Weak convergence ,Probability (math.PR) ,010102 general mathematics ,90C31 ,60F05, 60B12, 62E20 (Primary) 90C08, 90C31, 62G10 (Secondary) ,Delta method ,Metric space ,Statistics, Probability and Uncertainty ,60B12 ,Mathematics - Probability ,62G10 - Abstract
We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for non-linear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for ultra-metric spaces. Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.
- Published
- 2017
131. Invariance principles under the Maxwell–Woodroofe condition in Banach spaces
- Author
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Christophe Cuny, Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), and CentraleSupélec
- Subjects
Statistics and Probability ,Discrete mathematics ,Invariance principle ,Banach valued processes ,010102 general mathematics ,Banach space ,compact law of the iterated logarithm ,Law of the iterated logarithm ,Context (language use) ,Mathematical proof ,01 natural sciences ,Maxwell-Woodroofe's condition ,010104 statistics & probability ,Maxwell–Woodroofe’s condition ,Mathematics::Probability ,60F17 ,invariance principles ,37A50 ,[MATH]Mathematics [math] ,60F25 ,0101 mathematics ,Statistics, Probability and Uncertainty ,60B12 ,Random variable ,Mathematics - Abstract
We prove that, for (adapted) stationary processes, the so-called Maxwell–Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of $L^{p}$-valued random variables, with $1\le p
- Published
- 2017
132. On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size
- Author
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Koji Tsukuda
- Subjects
the Feller coupling ,Statistics and Probability ,Approximations of π ,Context (language use) ,Poisson distribution ,Combinatorics ,symbols.namesake ,Mathematics::Probability ,60F05 ,FOS: Mathematics ,Partition (number theory) ,Quantitative Biology::Populations and Evolution ,Central limit theorem ,Mathematics ,62E20 ,functional central limit theorem ,Probability (math.PR) ,Ewens sampling formula ,92D10 ,Sampling (statistics) ,Sample size determination ,Mutation (genetic algorithm) ,symbols ,Poisson approximation ,Statistics, Probability and Uncertainty ,60B12 ,Mathematics - Probability - Abstract
The Ewens sampling formula was firstly introduced in the context of population genetics by Warren John Ewens in 1972, and has appeared in a lot of other scientific fields. There are abundant approximation results associated with the Ewens sampling formula especially when one of the parameters, the sample size $n$ or the mutation parameter $\theta$ which denotes the scaled mutation rate, tends to infinity while the other is fixed. By contrast, the case that $\theta$ grows with $n$ has been considered in a relatively small number of works, although this asymptotic setup is also natural. In this paper, when $\theta$ grows with $n$, we advance the study concerning the asymptotic properties of the total number of alleles and of the counts of components in the allelic partition assuming the Ewens sampling formula from the viewpoint of Poisson approximations., Comment: 38 pages
- Published
- 2017
133. Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups
- Author
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Fabian Schwarzenberger, Christoph Schumacher, and Ivan Veselić
- Subjects
Statistics and Probability ,Pure mathematics ,amenable group ,Distribution (number theory) ,Uniform convergence ,FOS: Physical sciences ,Field (mathematics) ,Dynamical Systems (math.DS) ,Uniform norm ,60F99, 60B12, 62E20, 60K35 ,Følner sequence ,FOS: Mathematics ,Ergodic theory ,Mathematics - Dynamical Systems ,Mathematical Physics ,Mathematics ,62E20 ,Amenable group ,Mathematical Physics (math-ph) ,Empirical measures ,60K35 ,Bounded function ,quasi-tilings ,Statistics, Probability and Uncertainty ,Glivenko–Cantelli theory ,60F99 ,60B12 - Abstract
We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost additive, with finite range dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem, more precisely, that averages along a Foelner sequence converge uniformly to a limiting function. Moreover we give explicit error estimates for the approximation in the sup norm., Comment: 28 pages, 1 figure, accepted in Annales of Applied Probability
- Published
- 2017
- Full Text
- View/download PDF
134. Functional approximations with Stein's method of exchangeable pairs
- Author
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Mikołaj J. Kasprzak, Fonds National de la Recherche - FnR, and EPSRC [sponsor]
- Subjects
Statistics and Probability ,60B10 ,Probability (math.PR) ,Stein's method ,Stein’s method ,Exchangeable pairs ,Stochastic processes ,60F17 ,60B10, 60F17 (Primary) 60B12, 60J65, 60E05, 60E15 (Secondary) ,FOS: Mathematics ,Mathematics [G03] [Physical, chemical, mathematical & earth Sciences] ,60J65 ,Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre] ,60E05 ,Statistics, Probability and Uncertainty ,60E15 ,Humanities ,60B12 ,Mathematics - Probability ,Functional convergence ,Mathematics - Abstract
We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance of a sum of random variables, chosen from an array according to a random permutation, from a Gaussian mixture process. This result lets us prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the distribution of its rescaled edge counts to a continuous Gaussian process., Comment: will appear in Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques
- Published
- 2017
- Full Text
- View/download PDF
135. Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution
- Author
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Nicola Loperfido and Marcus C. Christiansen
- Subjects
Independent and identically distributed random variables ,Statistics and Probability ,Skew normal distribution ,General Mathematics ,Central limit theorem ,skewness ,third-order tensor ,Multivariate normal distribution ,skew normal ,010103 numerical & computational mathematics ,01 natural sciences ,15A69 ,Combinatorics ,order of convergence ,010104 statistics & probability ,60F05 ,Mixture distribution ,Applied mathematics ,Illustration of the central limit theorem ,Multivariate t-distribution ,0101 mathematics ,Mathematics ,Cramer's condition ,62E17 ,Matrix normal distribution ,Statistics, Probability and Uncertainty ,60B12 - Abstract
We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n -2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.
- Published
- 2014
136. Rigid representations of the multiplicative coalescent with linear deletion
- Author
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Balázs Ráth and James B. Martin
- Subjects
Erdős-Rényi random graph ,Statistics and Probability ,05C80 ,frozen percolation ,Markov process ,Lambda ,01 natural sciences ,Projection (linear algebra) ,Coalescent theory ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,multiplicative coalescent ,FOS: Mathematics ,0101 mathematics ,Connection (algebraic framework) ,Randomness ,Mathematics ,010102 general mathematics ,Multiplicative function ,Probability (math.PR) ,Random function ,symbols ,60J99 ,Statistics, Probability and Uncertainty ,60B12 ,Mathematics - Probability - Abstract
We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is deleted with rate $\lambda x$ (where $\lambda\geq 0$ is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case $\lambda=0$ (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a "tilt" of a random function, which increases with time; for $\lambda>0$ we find a novel representation in which this tilt is complemented by a "shift" mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are "rigid", in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes., Comment: 45 pages, 6 figures. Updated references
- Published
- 2016
137. Particle representations for stochastic partial differential equations with boundary conditions
- Author
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Christopher Janjigian, Dan Crisan, and Thomas G. Kurtz
- Subjects
interacting particle systems ,Statistics and Probability ,Statistics & Probability ,Boundary (topology) ,diffusions with reflecting boundary ,93E11 ,01 natural sciences ,Domain (mathematical analysis) ,010104 statistics & probability ,Stochastic differential equation ,symbols.namesake ,FOS: Mathematics ,35R60 ,Applied mathematics ,Boundary value problem ,60F25 ,0101 mathematics ,Mathematics ,stochastic Allen-Cahn equation ,Probability (math.PR) ,0104 Statistics ,010102 general mathematics ,Stochastic partial differential equations ,Function (mathematics) ,Stochastic partial differential equation ,Nonlinear system ,Euclidean quantum field theory equation with quartic interaction ,60F17 ,Dirichlet boundary condition ,60H15 ,symbols ,60H35 ,60H10 ,Statistics, Probability and Uncertainty ,60B12 ,Mathematics - Probability - Abstract
In this article, we study weighted particle representations for a class of stochastic partial differential equations (SPDE) with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations. The locations are given by independent, stationary reflecting diffusions in a bounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise $W$ which is the stochastic input for the SPDE. The weights interact through $V$, the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on the boundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure $V$ is the unique solution for each of the nonlinear stochastic partial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14, 15].
- Published
- 2016
138. On a memory game and preferential attachment graphs
- Author
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Hüseyin Acan and Pawel Hitczenko
- Subjects
Statistics and Probability ,memory game ,Convergence in distribution ,05C82 ,0102 computer and information sciences ,Expected value ,Preferential attachment ,Poisson distribution ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,Example of a game without a value ,60F05 ,FOS: Mathematics ,Mathematics - Combinatorics ,60C05 ,0101 mathematics ,preferential attachment graph ,generalized Pólya urn ,Mathematics ,Discrete mathematics ,Applied Mathematics ,Probability (math.PR) ,Connection (mathematics) ,Distribution (mathematics) ,Convergence of random variables ,010201 computation theory & mathematics ,Product (mathematics) ,symbols ,Combinatorics (math.CO) ,Primary 60F05, Secondary 05C82, 60B12, 60C05 ,60B12 ,Mathematics - Probability - Abstract
In a recent paper Velleman and Warrington analyzed the expected values of some of the parameters in a memory game, namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by--product of our methods we obtain simpler proofs (although without rate of convergence) of some of the results of Pek\"oz, R\"ollin, and Ross on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. For proving that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multi--type generalized P\'olya urn model., Comment: 24 pages, 1 figure
- Published
- 2016
139. Computation of instant system availability and its applications
- Author
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Patrick Kandege, Emmanuel Hagenimana, and Song Li-xin
- Subjects
Mathematical optimization ,Monotonicity ,Computation ,Value (computer science) ,Monotonic function ,010103 numerical & computational mathematics ,Interval (mathematics) ,01 natural sciences ,Upper and lower bounds ,90B25 ,26A48 ,010104 statistics & probability ,Renewal equation ,Instant availability ,0101 mathematics ,Repairable system ,Weibull distribution ,Mathematics ,Discrete mathematics ,Multidisciplinary ,Research ,Binary random variables ,Log-normal distribution ,37A60 ,11KXX ,60B12 ,Upper bound ,Instant - Abstract
The instant system availability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\tau (t)$$\end{document}Sτ(t) of a repairable system with the renewal equation was studied. The starting point monotonicity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\tau (t)$$\end{document}Sτ(t) was proved and the upper bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\tau (t)$$\end{document}Sτ(t) is also derived. It was found that the interval of instant system availability monotonically decreases. In addition, we provide examples that validate the analytically derived properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\tau (t)$$\end{document}Sτ(t) based on the Lognormal, Gamma and Weibull distributions and the results show that the value of T is slightly smaller than its value defined in Theorem 2. The procedure of using a bathtub as application for this article is also discussed.
- Published
- 2016
140. Functional valued pfh-order autoregressive processes
- Author
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Mourid, Tahar and HAL-SU, Gestionnaire
- Subjects
60G ,Probability on Banach spaces ,functional valued autoregressive processes ,46E15 ,60B12 ,[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST] ,covariance operator ,spectrum - Abstract
We present a class of function spaces valued pth-order autoregressive random processes and we study their main properties. We also give examples of special functional autoregressive representation (with not strong white noise) which is different of that given in Bosq [6]. This functional approach seems a promising tool to investigate statistical applications of continuous time random process and to obtain new limit theorems for dependent variables.
- Published
- 2016
141. Limiting distribution of elliptic homogenization error with periodic diffusion and random potential
- Author
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Wenjia Jing
- Subjects
Gaussian ,Asymptotic distribution ,weak convergence of probability distributions ,01 natural sciences ,Homogenization (chemistry) ,symbols.namesake ,Mathematics - Analysis of PDEs ,random field ,FOS: Mathematics ,35R60 ,0101 mathematics ,Mathematics ,Numerical Analysis ,Random field ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Hilbert space ,periodic and stochastic homogenization ,Differential operator ,probability measures on Hilbert space ,010101 applied mathematics ,Compact space ,symbols ,Probability distribution ,60B12 ,Analysis ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that in the short range correlation setting, the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by the homogenized Green's function, the homogenized solution and the statistics of the random potential. Similar results hold for random potentials that are functions of long range correlated Gaussian random fields. These results generalize previous ones in the setting with slowly varying diffusion coefficients, and the current setting with fast oscillations in the differential operator requires new methods to prove compactness of the probability distributions of the random fluctuation., 41 pages, 2 figures
- Published
- 2016
142. Théorèmes limites de la théorie des probabilités dans les systèmes dynamiques
- Author
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Giraudo, Davide, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Université de Rouen, Dalibor Volny (dalibor.volny@univ-rouen.fr), and Giraudo, Davide
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Mixing conditions ,Processus stationnaires ,Hölder spaces ,Théorème de la limite central ,Invariance principle ,Martingale difference sequences ,Critère projectif ,Sommes de variables aléatoires ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Probability inequalities ,Processus Mélangeants ,Principe d'invariance faible ,Champs aléatoires ,Orthomartingales ,Sums of random variables ,28D05 ,37A05 ,60B12 ,60F05 ,60F15 ,60F17 ,60G10 ,60G42 ,60G48 ,60G60 ,Stationary Processes ,Projective criterion - Abstract
This thesis is devoted to limit theorems for strictly stationary sequences and random fields. We concentrate essentially on the central limit theorem and its invariance principle.First, we show with the help of a counter-example that for a strictly stationary absolutely regular sequence, the central limit theorem may hold but not the invariance principle. We also show that the central limit theorem does not take place for partial sums of a Hilbert space valued, strictly stationary and absolutely regular sequence, even if we assume that the normalized partial sums form a uniformly integrable family. Second, we investigate the Holderian invariance principle. We treat the case of $\tau$-dependent (Dedecker, Prieur, 2005) and $\rho$-mixing strictly stationary sequences. We provide a sufficient condition on the law of a strictly stationary martingale difference sequence and the quadratic variance which guarantee the invariance principle in a Hölder space. We construct a counter-example which shows its sharpness. We derive conditions in the spirit of Hannan (1979), and Maxwell and Woodroofe (2000) by a martingale approximation.We then discuss the martingale/coboundary decomposition. In dimension one, we provide sharp integrability conditions on the transfer function and the coboundary for which the later does not spoil the invariance principle, the law of the iterated logarithm or the strong law of large numbers if these theorems take place for the martingale involved in the decomposition. We also provide a sufficient condition for an orthomartingale/coboundary decomposition for strictly stationary random fields.Lastly, we establish tails inequalities for orthomartingale and Bernoulli random fields. We prove an invariance principle in Hölder spaces for these random fields using such inequalities., Cette thèse est consacrée aux théorèmes limites pour les suites et les champs aléatoires strictement stationnaires. Nous étudions essentiellement le théorème limite central et sa version fonctionnelle, appelée principe d'invariance. Dans un premier temps, nous montrons à l'aide d'un contre-exemple que pour les processus strictement stationnaires $\beta$-mélangeants, le théorème limite central peut avoir lieu sans que ce ne soit le cas pour la version fonctionnelle. Nous montrons également que le théorème limite central n'a pas nécessairement lieu pour les sommes partielles d'une suite strictement stationnaire $\beta$-mélangeante à valeurs dans un espace de Hilbert de dimension infinie, même en supposant l'uniforme intégrabilité de la suite des sommes partielles normalisées.Puis nous étudions le principe d'invariance dans l'espace des fonctions hölderiennes. Nous traitons le cas des suites strictement stationnaires $\tau$-dépendantes (au sens de Dedecker, Prieur, 2005) ou $\rho$-mélangeantes. Nous donnons également une condition suffisante sur la loi d'une suite strictement stationnaire d'accroissements d'une martingale et la variance conditionnelle garantissant le principe d'invariance dans l'espace des fonctions hölderiennes, et nous démontrons son optimalité à l'aide d'un contre-exemple. Ensuite, nous déduisons grâce à une approximation par martingales des conditions dans l'esprit de celles de Hannan (1979), et Maxwell et Woodroofe (2000).Nous discutons ensuite de la décomposition martingale/cobord. Dans le cas des suites, nous fournissons des conditions d'intégrabilité sur la fonction de transfert et le cobord pour que ce dernier ne perturbe pas le principe d'invariance, la loi des logarithmes itérés ou bien la loi forte des grands nombres si ceux-ci ont lieu pour la martingale issue de la décomposition. Dans le cas des champs, nous formulons une condition suffisante pour une décomposition ortho-martingale/cobord. Enfin, nous établissons des inégalités sur les queues des maxima des sommes partielles d'un champ aléatoire de type ortho-martingale ou bien d'un champ qui s'exprime comme une fonctionnelle d'un champ i.i.d. Ces inégalités permettent d'obtenir un principe d'invariance dans les espaceshölderiens pour ces champs aléatoires.
- Published
- 2015
143. Convergence of bi-measure $\mathbb{R}$-trees and the pruning process
- Author
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Wolfgang Löhr, Guillaume Voisin, and Anita Winter
- Subjects
Statistics and Probability ,Prohorov metric ,Pointed Gromov-weak topology ,Measure (mathematics) ,05C05 ,Non-locally finite measures ,Mathematics::Probability ,60J25 ,60F05 ,CRT ,Pruning (decision trees) ,60G55 ,Statistics, Probability and Uncertainty ,Real trees ,Pruning procedure ,Tree-valued Markov process ,Humanities ,60B12 ,Mathematics ,05C10 - Abstract
Dans (Ann. Inst. Henri Poincare Probab. Stat. 34 (1998) 637–686), les auteurs obtiennent une chaine de Markov a valeurs arbres en elaguant de plus en plus de sous-arbres le long des nœuds d’un arbre de Galton–Watson. Plus recemment dans (Ann. Probab. 40 (2012) 1167–1211), un analogue continu de la dynamique d’elagage a valeurs arbres est construit sur des arbres de Levy. Dans cet article, nous presentons une nouvelle topologie qui permet de relier les dynamiques discretes et continues en les considerant comme des exemples du meme processus de Markov fort avec des conditions initiales differentes. Nous construisons ce processus d’elagage sur l’espace des arbres appeles bi-mesures, qui sont des espaces metriques mesures avec une mesure d’elagage additionnelle. La mesure d’elagage est supposee finie sur les arbres finis, mais pas necessairement localement finie. De plus, nous caracterisons analytiquement le processus d’elagage par son generateur infinitesimal et montrons qu’il est continu en son arbre bi-mesure initial. Plusieurs exemples sont donnes, notamment le cas d’une loi de reproduction a variance finie ou la mesure d’elagage est la mesure des longueurs sur l’arbre sous-jacent.
- Published
- 2015
144. Departure from normality of increasing-dimension martingales
- Author
-
Ignacio Arbués
- Subjects
Statistics and Probability ,media_common.quotation_subject ,Gaussian ,Residual autocorrelation ,Central limit theorem ,Banach space ,Asymptotic distribution ,Multivariate normal distribution ,symbols.namesake ,60F05 ,Statistics ,Applied mathematics ,Normality ,Mathematics ,media_common ,Numerical Analysis ,Confidence regions ,Banach spaces ,Approximate models ,Autoregressive model ,symbols ,62M10 ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,60B12 - Abstract
In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the k(n)-dimensional average of n martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of k(n) is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR(∞) and the order of the model grows with the length of the series.
- Published
- 2009
- Full Text
- View/download PDF
145. On an extension of the Hilbertian central limit theorem to Dirichlet forms
- Author
-
CHORRO christophe, Centre d'économie de la Sorbonne (CES), and Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Dirichlet forms ,49Q12 ,central limit theorem ,squared field operator ,31C25 ,31C25, 47B25, 49Q12, 60B12 ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,vectorial domain ,60F05 ,errors ,errors. " ,closability ,47B25 ,60B12 - Abstract
Permanent link to this document: http://projecteuclid.org/euclid.ojm/1216151109; International audience; In a recent paper, Nicolas Bouleau provides a new tool, based on the language of Dirichlet forms, to study the propagation of errors and reinforce the historical approach of Gauss. In the same way that the practical use of the normal distribution in statistics may be explained by the central limit theorem, the aim of this paper is to underline the importance of a family of error structures by asymptotic arguments.
- Published
- 2008
146. Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths
- Author
-
Yuzuru Inahama and Hiroshi Kawabi
- Subjects
Laplace approximations ,Rough path ,Series (mathematics) ,Laplace transform ,Mathematical analysis ,Space (mathematics) ,46T12 ,58J65 ,Asymptotic expansions ,Large deviation principle ,Laplace's method ,60G15 ,Large deviations theory ,Itô functional ,Rough path theory ,Laplace approximation ,Stochastic Taylor expansion ,60B12 ,Rate function ,Analysis ,Heat kernel ,stochastic Taylor expansions ,Mathematics - Abstract
In this paper, we establish asymptotic expansions for the Laplace approximations for Ito functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace's method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148–194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797–816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach space-valued Ito functionals of Brownian rough paths, in: Proceedings of the Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Ito, Springer, Berlin, in press. Available at: http://www.abelprisen.no/no/abelprisen/deltagere_2005.html ].
- Published
- 2007
147. A stochastic approximation approach to quasi-stationary distributions on finite spaces
- Author
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Bertrand Cloez, Michel Benaïm, Université de Neuchâtel (UNINE), Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme (MODEMIC), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Swiss National Foundation : FN 200020-149871/1, ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), ANR-11-IDEX-0002-02/11-LABX-0040,CIMI,Centre International de Mathématiques et d’Informatique (de Toulouse)(2011), Institut de Mathématiques (UNINE), and Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)
- Subjects
Statistics and Probability ,reinforced random walks ,Generalization ,walks ,Dynamical system ,Stochastic approximation ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,quasi-stationary distributions ,approximation method ,reinforced random ,random perturbations of dynamical ,Statistiques (Mathématiques) ,60J20 ,0101 mathematics ,random perturbations of dynamical systems ,Probability ,approximation stochastique ,Mathematics ,Particle system ,Simplex ,mathématique ,Probability (math.PR) ,010102 general mathematics ,Probabilités ,simulation ,16. Peace & justice ,65C20, 60B12, 60J10 ,34F05, 60J20 ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,34F05 ,probabilité ,système dynamique ,60J10 ,65C20 ,Vector field ,Statistics, Probability and Uncertainty ,60B12 ,algorithme ,Mathematics - Probability - Abstract
Swiss National Foundation : FN 200020-149871/1. CIMI (Centre International de Mathematiques et d'Informatique) : ANR-11-LABX-0040-CIMI, ANR-11-IDEX-0002-02; International audience; This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a method introduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise asymptotic rates for this type of algorithm. In the last part, our convergence results are compared with those of a particle system algorithm (a discrete-time version of the Fleming-Viot algorithm).
- Published
- 2015
148. Large deviation for stochastic line integrals as Lp-currents
- Author
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Kusuoka, Shigeo, Kuwada, Kazumasa, and Tamura, Yozo
- Published
- 2010
- Full Text
- View/download PDF
149. On limit theorems for Banach-space-valued linear processes
- Author
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Račkauskas, A. and Suquet, Ch.
- Published
- 2010
- Full Text
- View/download PDF
150. Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces
- Author
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Van Dung, Le, Ngamkham, Thuntida, Tien, Nguyen Duy, and Volodin, A. I.
- Published
- 2009
- Full Text
- View/download PDF
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