Probability Theory and Related Fields, ISSN 0178-8051, 2/2016, Volume 164, Issue 1, pp. 61 - 108

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process...

62G05 | Secondary 60G51 | Primary 60F05 | Empirical process | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Lévy process | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Donsker theorem | High-frequency inference | Statistics for Business/Economics/Mathematical Finance/Insurance | Operation Research/Decision Theory | CENTRAL LIMIT-THEOREMS | RATES | NONPARAMETRIC-ESTIMATION | Levy process | CONVERGENCE | STATISTICS & PROBABILITY | INFERENCE | Studies | Mathematical analysis | Brownian movements | Asymptotic methods | Brownian motion | Theorems | Infinity | Asymptotic properties | Gaussian | Sampling | Estimators | Constraining

62G05 | Secondary 60G51 | Primary 60F05 | Empirical process | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Lévy process | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Donsker theorem | High-frequency inference | Statistics for Business/Economics/Mathematical Finance/Insurance | Operation Research/Decision Theory | CENTRAL LIMIT-THEOREMS | RATES | NONPARAMETRIC-ESTIMATION | Levy process | CONVERGENCE | STATISTICS & PROBABILITY | INFERENCE | Studies | Mathematical analysis | Brownian movements | Asymptotic methods | Brownian motion | Theorems | Infinity | Asymptotic properties | Gaussian | Sampling | Estimators | Constraining

Journal Article

The Annals of probability, ISSN 0091-1798, 7/2003, Volume 31, Issue 3, pp. 1228 - 1240

.... In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law...

Mathematical theorems | Central limit theorem | Statistical theories | Probability theory | Partial sums | Mathematical moments | Random variables | Probabilities | Law of sines | Self-normalized sums | Arc sine law | Donsker's theorem | arc sine law | STATISTICS & PROBABILITY | self-normalized sums | 60F17 | 60F05 | 62E20

Mathematical theorems | Central limit theorem | Statistical theories | Probability theory | Partial sums | Mathematical moments | Random variables | Probabilities | Law of sines | Self-normalized sums | Arc sine law | Donsker's theorem | arc sine law | STATISTICS & PROBABILITY | self-normalized sums | 60F17 | 60F05 | 62E20

Journal Article

ELECTRONIC COMMUNICATIONS IN PROBABILITY, ISSN 1083-589X, 2020, Volume 25

We compute the Wassertein-1 (or Kantorovitch-Rubinstein) distance between a random walk in R-d and the Brownian motion. The proof is based on a new estimate of...

STEINS METHOD | Wasserstein distance | STATISTICS & PROBABILITY | Malliavin calculus | Stein's method | Donsker theorem

STEINS METHOD | Wasserstein distance | STATISTICS & PROBABILITY | Malliavin calculus | Stein's method | Donsker theorem

Journal Article

Statistics and Probability Letters, ISSN 0167-7152, 07/2019, Volume 150, pp. 1 - 8

In this paper we study a Donsker type theorem for the fractional Poisson process (fPp...

Fractional Poisson process | Donsker type theorem | Skorohod topology | Long memory | Random walk | LEVY PROCESSES | STATISTICS & PROBABILITY | PARAMETERS | ROSENBLATT PROCESS | Memory

Fractional Poisson process | Donsker type theorem | Skorohod topology | Long memory | Random walk | LEVY PROCESSES | STATISTICS & PROBABILITY | PARAMETERS | ROSENBLATT PROCESS | Memory

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2012, Volume 55, Issue 11, pp. 2285 - 2296

... Berlin Heidelberg 2012 math.scichina.com www.springerlink.com Stochastic quantization and ergodic theorem for density of diﬀusions HU YaoZhong Department...

微积分 | 扩散过程 | 量化 | 遍历定理 | 概率密度函数 | 随机 | 改造技术 | 融合率 | 60H10 | Stochastic quantization | Donsker delta functional | Mathematics | diffusions | Malliavan calculus | Girsanov formula | 37A30 | 81S20 | ergodic theorem | heat kernel | Clark-Ocone formula | Applications of Mathematics | local time | 60J55 | MATHEMATICS, APPLIED | TIME | MATHEMATICS | HEAT KERNELS | Mathematical analysis | China | Transformations | Calculus | Diffusion | Density | Probability density functions | Convergence | Series (mathematics) | Texts | Astronomy

微积分 | 扩散过程 | 量化 | 遍历定理 | 概率密度函数 | 随机 | 改造技术 | 融合率 | 60H10 | Stochastic quantization | Donsker delta functional | Mathematics | diffusions | Malliavan calculus | Girsanov formula | 37A30 | 81S20 | ergodic theorem | heat kernel | Clark-Ocone formula | Applications of Mathematics | local time | 60J55 | MATHEMATICS, APPLIED | TIME | MATHEMATICS | HEAT KERNELS | Mathematical analysis | China | Transformations | Calculus | Diffusion | Density | Probability density functions | Convergence | Series (mathematics) | Texts | Astronomy

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 7/2008, Volume 141, Issue 3, pp. 333 - 387

...). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included. [PUBLICATION ABSTRACT]

Uniform central limit theorem | Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Mathematical Biology in General | Smoothed empirical processes | Secondary: 60F05 | Kernel density estimation | Plug-in property | Primary: 62G07 | uniform central limit theorem | WEAK-CONVERGENCE | plug-in property | DONSKER CLASSES | STATISTICS & PROBABILITY | smoothed empirical processes | Studies | Data smoothing | Mathematical models

Uniform central limit theorem | Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Mathematical Biology in General | Smoothed empirical processes | Secondary: 60F05 | Kernel density estimation | Plug-in property | Primary: 62G07 | uniform central limit theorem | WEAK-CONVERGENCE | plug-in property | DONSKER CLASSES | STATISTICS & PROBABILITY | smoothed empirical processes | Studies | Data smoothing | Mathematical models

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2019, Volume 477, Issue 2, pp. 1133 - 1156

... convex functionals on Wasserstein-1. We retrieve dual transportation inequalities as a Corollary and we provide examples where the theorem can be used to easily prove dual expressions like the celebrated Donsker-Varadhan variational formula...

Fenchel-Moreau-Rockafellar theorem | Conjugate duality | Donsker-Varadhan variational formula | Optimal control | Wasserstein metric | Partially observable Markov decision processes | Markov processes | Neurosciences | Mathematics - Probability

Fenchel-Moreau-Rockafellar theorem | Conjugate duality | Donsker-Varadhan variational formula | Optimal control | Wasserstein metric | Partially observable Markov decision processes | Markov processes | Neurosciences | Mathematics - Probability

Journal Article

The Annals of probability, ISSN 0091-1798, 7/2005, Volume 33, Issue 4, pp. 1422 - 1451

.... By the central limit theorem for diffusions, the finite-dimensional distributions of G converge weakly to those of a zero-mean Gaussian random process G...

Ergodic theory | Sufficient conditions | Speed | Maps | Central limit theorem | Mathematical theorems | Covariance | Entropy | Martingales | Perceptron convergence procedure | Majorizing measures | Continuous martingales | Uniform central limit theorem | Diffusions | Local time estimator | Local time | Donsker class | local time estimator | uniform central limit theorem | majorizing measures | ERGODIC DIFFUSION | STATISTICS & PROBABILITY | diffusions | continuous martingales | EFFICIENCY | local time | Mathematics - Probability | 60J60 | 62M05 | 60F17 | 60J55

Ergodic theory | Sufficient conditions | Speed | Maps | Central limit theorem | Mathematical theorems | Covariance | Entropy | Martingales | Perceptron convergence procedure | Majorizing measures | Continuous martingales | Uniform central limit theorem | Diffusions | Local time estimator | Local time | Donsker class | local time estimator | uniform central limit theorem | majorizing measures | ERGODIC DIFFUSION | STATISTICS & PROBABILITY | diffusions | continuous martingales | EFFICIENCY | local time | Mathematics - Probability | 60J60 | 62M05 | 60F17 | 60J55

Journal Article

The Annals of probability, ISSN 0091-1798, 7/2015, Volume 43, Issue 4, pp. 1777 - 1822

.... It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach's fixed-point theorem...

Functional limit theorem | Zolotarev metric | Recursive distributional equation | Contraction method | Donsker's invariance principle | PARTIAL MATCH QUERIES | TREES | QUICKSORT | LIMIT-THEOREM | RANDOM QUADTREES | contraction method | STATISTICS & PROBABILITY | RECURSIVE ALGORITHMS | recursive distributional equation | 68Q25 | 60C05 | Donsker’s invariance principle | 60G18 | 60F17

Functional limit theorem | Zolotarev metric | Recursive distributional equation | Contraction method | Donsker's invariance principle | PARTIAL MATCH QUERIES | TREES | QUICKSORT | LIMIT-THEOREM | RANDOM QUADTREES | contraction method | STATISTICS & PROBABILITY | RECURSIVE ALGORITHMS | recursive distributional equation | 68Q25 | 60C05 | Donsker’s invariance principle | 60G18 | 60F17

Journal Article

Mathematics of Operations Research, ISSN 0364-765X, 02/2007, Volume 32, Issue 1, pp. 118 - 135

We consider empirical approximations (sample average approximations) of two-stage stochastic mixed-integer linear programs and derive central limit theorems for the objectives and optimal values...

Hadamard directional differentiability | delta method | stochastic programming | empirical process | subsampling | mixed-integer optimization | bootstrap | sample average approximation | stability | Donsker class | Integer programming | Integers | Mathematical intervals | Mathematical theorems | Approximation | Mathematics | Mathematical functions | Statistics | Confidence interval | Bootstrap resampling | Empirical process | Stability | Bootstrap | Delta method | Subsampling | Mixed-integer optimization | Sample average approximation | Stochastic programming | MATHEMATICS, APPLIED | RECOURSE | MAXIMUM-LIKELIHOOD ESTIMATORS | VON-MISES METHOD | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BOUNDS | STATISTICAL ESTIMATORS | GLIVENKO-CANTELLI PROBLEM | CONVERGENCE | OPTIMIZATION | Confidence intervals | Studies | Theorems | Stochastic models

Hadamard directional differentiability | delta method | stochastic programming | empirical process | subsampling | mixed-integer optimization | bootstrap | sample average approximation | stability | Donsker class | Integer programming | Integers | Mathematical intervals | Mathematical theorems | Approximation | Mathematics | Mathematical functions | Statistics | Confidence interval | Bootstrap resampling | Empirical process | Stability | Bootstrap | Delta method | Subsampling | Mixed-integer optimization | Sample average approximation | Stochastic programming | MATHEMATICS, APPLIED | RECOURSE | MAXIMUM-LIKELIHOOD ESTIMATORS | VON-MISES METHOD | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BOUNDS | STATISTICAL ESTIMATORS | GLIVENKO-CANTELLI PROBLEM | CONVERGENCE | OPTIMIZATION | Confidence intervals | Studies | Theorems | Stochastic models

Journal Article

International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, ISSN 0218-4885, 02/2018, Volume 26, Issue 1, pp. 27 - 42

In this note - starting from d-dimensional ( with d > 1) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk...

Brownian motion | Donsker's theorem | d -dimensional fuzzy vectors | fuzzy random walk | support function | EXISTENCE | STOCHASTIC DIFFERENTIAL-EQUATIONS | METRIC-SPACES | NON-LIPSCHITZ COEFFICIENTS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | RANDOM-VARIABLES | UNIQUENESS | GAUSSIAN-PROCESSES | LIMIT-THEOREMS | SETS | d-dimensional fuzzy vectors | LOCAL MARTINGALES

Brownian motion | Donsker's theorem | d -dimensional fuzzy vectors | fuzzy random walk | support function | EXISTENCE | STOCHASTIC DIFFERENTIAL-EQUATIONS | METRIC-SPACES | NON-LIPSCHITZ COEFFICIENTS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | RANDOM-VARIABLES | UNIQUENESS | GAUSSIAN-PROCESSES | LIMIT-THEOREMS | SETS | d-dimensional fuzzy vectors | LOCAL MARTINGALES

Journal Article

Modern Stochastics: Theory and Applications, ISSN 2351-6046, 09/2017, Volume 4, Issue 3, pp. 189 - 198

...])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Cs\"{o}rg\H{o} et al. (2003). Some notes on spheres with respect...

Mathematics - Probability | Brownian motion | Poincaré–Borel lemma | Donsker theorem | self-normalized sums

Mathematics - Probability | Brownian motion | Poincaré–Borel lemma | Donsker theorem | self-normalized sums

Journal Article

Journal of theoretical probability, ISSN 1572-9230, 2007, Volume 20, Issue 2, pp. 177 - 199

...J Theor Probab (2007) 20: 177–199 DOI 10.1007/s10959-007-0058-1 Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes...

Uniform Donsker class | Uniform metric entropy | Metric entropy with bracketing | Probability Theory and Stochastic Processes | Glivenko-Cantelli | Mathematics | Statistics, general | Sobolev, Besov, Hölder, Triebel, and Bessel potential spaces | P-VARIATION | LAW | NUMBERS | WEIGHTED FUNCTION-SPACES | STATISTICS & PROBABILITY | ITERATED LOGARITHM | PROBABILITY-INEQUALITIES | Sobolev, Besov, Holder, Triebel, and Bessel potential spaces | uniform Donsker class | uniform metric entropy | metric entropy with bracketing | ESTIMATORS | CONVERGENCE | UNIFORM DONSKER CLASSES | OPERATORS

Uniform Donsker class | Uniform metric entropy | Metric entropy with bracketing | Probability Theory and Stochastic Processes | Glivenko-Cantelli | Mathematics | Statistics, general | Sobolev, Besov, Hölder, Triebel, and Bessel potential spaces | P-VARIATION | LAW | NUMBERS | WEIGHTED FUNCTION-SPACES | STATISTICS & PROBABILITY | ITERATED LOGARITHM | PROBABILITY-INEQUALITIES | Sobolev, Besov, Holder, Triebel, and Bessel potential spaces | uniform Donsker class | uniform metric entropy | metric entropy with bracketing | ESTIMATORS | CONVERGENCE | UNIFORM DONSKER CLASSES | OPERATORS

Journal Article

Alea, ISSN 1980-0436, 2015, Volume 12, Issue 1, pp. 1 - 23

We consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with...

Donsker-varadhan theorem | Phonon boltzmann equation | Empirical flow | Large deviations | PROCESS EXPECTATIONS | LARGE TIME | THEOREM | Phonon Boltzmann equation | DYNAMICS | STATISTICS & PROBABILITY | Donsker-Varadhan theorem | EQUATION

Donsker-varadhan theorem | Phonon boltzmann equation | Empirical flow | Large deviations | PROCESS EXPECTATIONS | LARGE TIME | THEOREM | Phonon Boltzmann equation | DYNAMICS | STATISTICS & PROBABILITY | Donsker-Varadhan theorem | EQUATION

Journal Article

Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, 2014, Volume 19, Issue 6, pp. 1549 - 1562

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can...

Complete market | Incomplete market | Binomial approximation | Discretzation of Ito equations | Stochastic processes | Donsker theorem | binomial approximation | discretzation of Ito equations | MATHEMATICS, APPLIED | MARKET | STATISTICS | THEOREMS | EQUATIONS | complete market | MODEL | incomplete market

Complete market | Incomplete market | Binomial approximation | Discretzation of Ito equations | Stochastic processes | Donsker theorem | binomial approximation | discretzation of Ito equations | MATHEMATICS, APPLIED | MARKET | STATISTICS | THEOREMS | EQUATIONS | complete market | MODEL | incomplete market

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 03/2005, Volume 131, Issue 3, pp. 442 - 458

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established...

Permanent process | Donsker’s theorem | Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Invariance principle | Mathematics | Quantitative Finance | Elementary symmetric polynomial process | Statistics for Business/Economics/Mathematical Finance/Insurance | Random permanent | Multiple stochastic integral | Mathematical Biology in General | Orthogonal decomposition | Operation Research/Decision Theory | Donsker's theorem | random permanent | elementary symmetric polynomial process | MATRIX | ELEMENTARY SYMMETRIC POLYNOMIALS | LIMIT-THEOREMS | BEHAVIOR | permanent process | invariance principle | orthogonal decomposition | STATISTICS & PROBABILITY | multiple stochastic integral

Permanent process | Donsker’s theorem | Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Invariance principle | Mathematics | Quantitative Finance | Elementary symmetric polynomial process | Statistics for Business/Economics/Mathematical Finance/Insurance | Random permanent | Multiple stochastic integral | Mathematical Biology in General | Orthogonal decomposition | Operation Research/Decision Theory | Donsker's theorem | random permanent | elementary symmetric polynomial process | MATRIX | ELEMENTARY SYMMETRIC POLYNOMIALS | LIMIT-THEOREMS | BEHAVIOR | permanent process | invariance principle | orthogonal decomposition | STATISTICS & PROBABILITY | multiple stochastic integral

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 12/2016, Volume 29, Issue 4, pp. 1458 - 1484

.... For the case of compact domain, Chang has built upon a nonlinear version of the Krein–Rutman theorem to give a ‘min–max...

93E20 | Risk-sensitive control | Nisio semigroup | Secondary 60F10 | Probability Theory and Stochastic Processes | Principal eigenvalue | Mathematics | Statistics, general | Collatz–Wielandt formula | Donsker–Varadhan functional | Primary 60J60 | Variational formulation | Collatz-Wielandt formula | Donsker-Varadhan functional | STATISTICS & PROBABILITY | KREIN-RUTMAN THEOREM | Electrical engineering | Electric properties | Mathematics - Optimization and Control

93E20 | Risk-sensitive control | Nisio semigroup | Secondary 60F10 | Probability Theory and Stochastic Processes | Principal eigenvalue | Mathematics | Statistics, general | Collatz–Wielandt formula | Donsker–Varadhan functional | Primary 60J60 | Variational formulation | Collatz-Wielandt formula | Donsker-Varadhan functional | STATISTICS & PROBABILITY | KREIN-RUTMAN THEOREM | Electrical engineering | Electric properties | Mathematics - Optimization and Control

Journal Article

Markov Processes and Related Fields, ISSN 1024-2953, 2014, Volume 20, Issue 1, pp. 167 - 172

... among Gaussian processes. In this note, making use of the Mallows distance and for 1 < alpha <= 2, we derive the classical Donsker's theorem for alpha-stable Levy motions.

Stable laws | Donsker's theorem | Mallows distance | CONVERGENCE | STATISTICS & PROBABILITY | stable laws | LAWS

Stable laws | Donsker's theorem | Mallows distance | CONVERGENCE | STATISTICS & PROBABILITY | stable laws | LAWS

Journal Article

Statistics and Probability Letters, ISSN 0167-7152, 02/2013, Volume 83, Issue 2, pp. 588 - 595

.... The limit in law is a certain continuous state-space branching process (CSBP). In the present work, we obtain a Central Limit Theorem, thus completing Bertoin’s work.

Branching process | Lévy-Itô decomposition | Donsker’s invariance principle | Skorohod’s representation | Skorohod's representation | Donsker's invariance principle | Lévy-ItÔ decomposition | STATISTICS & PROBABILITY | Levy-Ito decomposition | Probability | Mathematics

Branching process | Lévy-Itô decomposition | Donsker’s invariance principle | Skorohod’s representation | Skorohod's representation | Donsker's invariance principle | Lévy-ItÔ decomposition | STATISTICS & PROBABILITY | Levy-Ito decomposition | Probability | Mathematics

Journal Article

Electronic Journal of Statistics, ISSN 1935-7524, 2012, Volume 6, pp. 2486 - 2518

.... We obtain a uniform central limit theorem with root nrate on the assumption that the smoothness of the functionals is larger than the illposedness of the problem, which is given by the polynomial...

Distribution function | Fourier multipliers | Deconvolution | Smoothed empirical processes | Efficiency | Donsker theorem | RATES | efficiency | LINEAR FUNCTIONALS | STATISTICS & PROBABILITY | smoothed empirical processes | ASYMPTOTIC NORMALITY | distribution function

Distribution function | Fourier multipliers | Deconvolution | Smoothed empirical processes | Efficiency | Donsker theorem | RATES | efficiency | LINEAR FUNCTIONALS | STATISTICS & PROBABILITY | smoothed empirical processes | ASYMPTOTIC NORMALITY | distribution function

Journal Article

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