Search results for "Convergence of random variables"

showing 6 items of 6 documents

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

2021

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see t…

Statistics and ProbabilityPhysicsAlgebra and Number TheorySpectral power distributionComputer Science::Information RetrievalProbability (math.PR)Astrophysics::Instrumentation and Methods for AstrophysicsBlock (permutation group theory)Marchenko–Pastur lawComputer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing)Bilinear form60F05 60B20 47N30Sample mean and sample covarianceCombinatoricsConvergence of random variablesFOS: Mathematicssample covariance matricesComputer Science::General LiteratureDiscrete Mathematics and CombinatoricsRandom matriceshigh dimensional statisticsStatistics Probability and UncertaintyRandom matrixRandom variableMathematics - ProbabilityRandom Matrices: Theory and Applications
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On almost sure convergence of amarts and martingales without the Radon-Nikodym property

1988

It is shown here that for any Banach spaceE-valued amart (X n) of classB, almost sure convergence off(Xn) tof(X) for eachf in a total subset ofE * implies scalar convergence toX.

Statistics and ProbabilityRadon–Nikodym theoremDiscrete mathematicsPure mathematicsConvergence of random variablesGeneral MathematicsScalar (mathematics)Statistics Probability and UncertaintyMathematicsJournal of Theoretical Probability
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Recursive estimation of the conditional geometric median in Hilbert spaces

2012

International audience; A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted L1 criterion and is consequently well adapted for robust online estimation. The weights are controlled by a kernel function and an associated bandwidth. Almost sure convergence and L2 rates of convergence are proved under general conditions on the conditional distribution as well as the sequence of descent steps of the algorithm and the sequence of bandwidths. Asymptotic normality is also proved for the averaged version of the algorithm with an optimal rate of convergence. A simulation study confirm…

Statistics and ProbabilityMallows-Wasserstein distanceRobbins-Monroasymptotic normalityCLTcentral limit theoremAsymptotic distributionMathematics - Statistics TheoryStatistics Theory (math.ST)01 natural sciencesMallows–Wasserstein distanceonline data010104 statistics & probability[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]60F05FOS: MathematicsApplied mathematics[ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST]0101 mathematics62L20MathematicsaveragingSequential estimation010102 general mathematicsEstimatorRobbins–MonroConditional probability distribution[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]Geometric medianstochastic gradient[ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH]robust estimatorRate of convergenceConvergence of random variablesStochastic gradient.kernel regressionsequential estimationKernel regressionStatistics Probability and Uncertainty
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Stochastic order characterization of uniform integrability and tightness

2013

We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. Especially, we show that whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics.

Statistics and ProbabilityDiscrete mathematicsPure mathematicsRandom fieldMultivariate random variableProbability (math.PR)ta111Random functionRandom element60E15 60B10 60F25Stochastic orderingFunctional Analysis (math.FA)Mathematics - Functional AnalysisRandom variateConvergence of random variablesStochastic simulationFOS: MathematicsStatistics Probability and UncertaintyMathematics - ProbabilityMathematicsStatistics & Probability Letters
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Variable Length Memory Chains: Characterization of stationary probability measures

2021

Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, turn out to be an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of the existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a general VLMC to admit a unique invariant probability measure. This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enj…

Statistics and ProbabilityPure mathematicsLongest Internal SuffixStationary distributionMarkov chain60J05 60C05 60G10Probability (math.PR)010102 general mathematics01 natural sciencesMeasure (mathematics)Variable Length Memory Chains010104 statistics & probabilityProbability theoryConvergence of random variablesFOS: MathematicsCountable setState spaceRenewal theory[MATH]Mathematics [math]0101 mathematicsstable context treessemi-Markov chainsMathematics - Probabilitystationary probability measureMathematicsBernoulli
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Asymptotic optimality of myopic information-based strategies for Bayesian adaptive estimation

2016

This paper presents a general asymptotic theory of sequential Bayesian estimation giving results for the strongest, almost sure convergence. We show that under certain smoothness conditions on the probability model, the greedy information gain maximization algorithm for adaptive Bayesian estimation is asymptotically optimal in the sense that the determinant of the posterior covariance in a certain neighborhood of the true parameter value is asymptotically minimal. Using this result, we also obtain an asymptotic expression for the posterior entropy based on a novel definition of almost sure convergence on "most trials" (meaning that the convergence holds on a fraction of trials that converge…

Statistics and ProbabilityAsymptotic analysisMathematical optimizationPosterior probabilityBayesian probabilityMathematics - Statistics TheoryStatistics Theory (math.ST)050105 experimental psychologydifferential entropyDifferential entropyactive data selection03 medical and health sciences0302 clinical medicineactive learningFOS: Mathematics0501 psychology and cognitive sciencescost of observationdecision theoryMathematicsD-optimalityBayes estimatorSequential estimation05 social sciencesBayesian adaptive estimationAsymptotically optimal algorithmConvergence of random variablesasymptotic optimalitysequential estimation030217 neurology & neurosurgery
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