Probability Theory

Klenke, Achim

doi:10.1007/978-3-030-56402-5

Cited by 55 publications

(34 citation statements)

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“…1 and γ(a) = (c s ) |s|>a/2 1 , which converges to 0 as a → ∞. For (S1), note that lim a→∞ a k γ(a) = 0 for all k ∈ N since (10). (S2), (S3) hold trivially, and for (S4) we can choose a n .…”

Section: Ma(∞)-processesmentioning

confidence: 96%

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The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method

Fleermann¹,

Kirsch²

2022

Preprint

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In this paper, we derive a central limit theorem for collections of weakly correlated random variables indexed by discrete metric spaces, where the correlation decays in the distance of the indices. The correlation structure we study depends solely on the separability of mixed moments. Our investigation yields a new proof for the CLT for α-mixing random variables, but also non-α-mixing random variables fit within our framework, such as MA(∞) processes. In particular, our results can be applied to ARMA(p, q) process with independent white noise.

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Section: Ma(∞)-processesmentioning

confidence: 96%

“…This has to do with the fact that there are also substantially weaker mixing concepts in ergodic theory called "mixing" and "ergodic". The connection is that α-mixing implies mixing, which implies ergodic, see [10] for details.…”

Section: Mixing Random Variablesmentioning

confidence: 99%

The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method

Fleermann¹,

Kirsch²

2022

Preprint

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“…For more background on conditional expectation and general disintegration we refer to [19,23]. We call a copula C ∈ C d completely dependent (w.r.t.…”

Section: S Will Be Denoted By B(s)mentioning

confidence: 99%

“…Consider a sequence of copulas (C n ) n∈N in C 3 c and an absolutely continuous copula C ∈ C 3 c , and let c 13 , c 23 denote the densities of the marginal copulas C 13 , C 23 of C, respectively. If there exist some constants p 13 , p 23…”

Section: Convergence With Respect To Dmentioning

confidence: 99%

How simplifying and flexible is the simplifying assumption in pair-copula constructions – analytic answers in dimension three and a glimpse beyond

Mroz

Fuchs

Trutschnig

2021

Electron. J. Statist.

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Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used 'simplifying assumption' is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional copulas with respect to the uniform metric d∞ -considering stronger notions of convergence like the one induced by the metric D 1 , by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turns out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to d∞ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula C, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28% of the diameter of the metric space. Moreover, the mapping ψ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to d∞ (but continuous with respect to D 1 and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning d∞ are then extended to the general multivariate setting.

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“…In the following, we introduce the necessary notation from probability theory [18]. Let (Ω, A, P ) be a probability space.…”

Section: Preliminaries In Probability Theorymentioning

confidence: 99%

On the Dynamical System of Principal Curves in $\mathbb R^d$

Beinert,

Bërdëllima,

Gräf

et al. 2021

Preprint

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Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized-from a stochastic point of view-as so-called self-consistent curves based on the conditional expectation andfrom the variational-calculus point of view-as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to R d with d ≥ 3. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in R 3 .

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Probability Theory

Cited by 55 publications

References 0 publications

The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method

The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method

How simplifying and flexible is the simplifying assumption in pair-copula constructions – analytic answers in dimension three and a glimpse beyond

On the Dynamical System of Principal Curves in $\mathbb R^d$

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