Orthogonal series and limit theorems for canonical U-and V-statistics of stationary connected observations

Borisov, I. S.; Volodko, N. V.

doi:10.3103/s1055134408040020

Cited by 16 publications

(23 citation statements)

References 5 publications

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“…The last assertion under the assumption n≥0 ϕ(n) 1/2 < ∞ is, up to inessential details, Theorem 5 in [26]. In [9] the authors express their doubts on correctness in [26] to substituting a dependent process into the function (50). Our conclusion agrees with that of [26].…”

Section: Exemplary Applicationssupporting

confidence: 67%

“…Using this relation along with (15) and (18) The following sufficient condition for convergence of the series in (14) will be used in Section 9 when considering applications. Expansion of a kernel into an absolutely convergent series whose summands are products of functions in separate variables is natural in the context of the limit theory of U-and V -statistics (see, for example, [9]). Projective tensor products call for using such series to representing arbitrary elements (see Proposition 2.8 in [49]).…”

Section: Growth Rates For Multiparameter Sumsmentioning

confidence: 99%

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Limit theorems for von Mises statistics of a measure preserving transformation

Denker

Gordin

2013

Probab. Theory Relat. Fields

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For a measure preserving transformation T of a probability space (X, F , µ) and some d ≥ 1 we investigate almost sure and distributional convergence of random variables of the formwhere C 1 , C 2 , . . . are normalizing constants and the kernel f belongs to an appropriate subspace in some L p (X d , F ⊗d , µ d ). We establish a form of the individual ergodic theorem for such sequences.Using a filtration compatible with T and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and d = 2, we also show that the convergence holds in distribution towards a quadratic form ∞ m=1 λ m η 2 m in independent standard Gaussian variables η 1 , η 2

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Section: Exemplary Applicationssupporting

confidence: 67%

Section: Growth Rates For Multiparameter Sumsmentioning

confidence: 99%

Limit theorems for von Mises statistics of a measure preserving transformation

Denker

Gordin

2013

Probab. Theory Relat. Fields

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“…However, it is not guaranteed that the joint distribution of X t and X t+h is absolutely continuous with respect to P X 0 ⊗ P X 0 , that is, (1) might fail on a set with nonzero measure. Borisov and Volodko (2008) have shown that this problem does not appear if an additional smoothness assumption on the kernel h is imposed what we therefore also do here. Furthermore, while a proof of the fact that lim sup…”

Section: Remark 1 (I)mentioning

confidence: 63%

“…In the case of dependent random variables, however, this approach requires some care. As pointed out by Borisov and Volodko (2008), approximation (1) is valid for almost all (x , y ) ∈ supp(P X 0 ⊗ P X 0 ). However, it is not guaranteed that the joint distribution of X t and X t+h is absolutely continuous with respect to P X 0 ⊗ P X 0 , that is, (1) might fail on a set with nonzero measure.…”

Section: Remark 1 (I)mentioning

confidence: 88%

“…Gregory (1977) and Serfling (1980). This method has been adopted for mixing random variables by Eagleson (1979), Carlstein (1988 and Borisov and Volodko (2008) as well as for associated random variables by Dewan and Prakasa Rao (2001) and Huang and Zhang (2006). In the case of dependent random variables, however, this approach requires some care.…”

Section: Remark 1 (I)mentioning

confidence: 99%

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Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics

Leucht

Neumann

2012

Ann Inst Stat Math

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We derive the asymptotic distributions of degenerate U -and V -statistics of stationary and ergodic random variables. Statistics of these types naturally appear as approximations of test statistics. Since the limit variables are of complicated structure, typically depending on unknown parameters, quantiles can hardly be obtained directly. Therefore, we prove a general result on the consistency of model-based bootstrap methods for U -and V -statistics under easily verifiable conditions. Three applications to hypothesis testing are presented. Finally, the finite sample behavior of the bootstrapbased tests is illustrated by a simulation study.

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$$U$$ U -Statistics of Ornstein–Uhlenbeck Branching Particle System

Adamczak

Miłoś

2013

J Theor Probab

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We consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R d and undergoing a binary, supercritical branching with a constant rate λ > 0.This system is known to fulfil a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem has been addressed. It turns out that the normalization and form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the Orstein-Uhlenbeck process. In the present paper we extend those results to U -statistics of the system proving a law of large numbers and a central limit theorem. MSC: primary 60F05; 60J80 secondary 60G20

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Orthogonal series and limit theorems for canonical U-and V-statistics of stationary connected observations

Cited by 16 publications

References 5 publications

Limit theorems for von Mises statistics of a measure preserving transformation

Limit theorems for von Mises statistics of a measure preserving transformation

Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics

$$U$$ U -Statistics of Ornstein–Uhlenbeck Branching Particle System

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