Convergence to a Stable Distribution Via Order Statistics

LePage, Raoul; Woodroofe, Michael; Zinn, Joel

doi:10.1214/aop/1176994367

Cited by 173 publications

(113 citation statements)

References 5 publications

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“…We can now state the representation which we will use. This result follows from [9] (for more details, see [10]). More generally, it is known (cf.…”

mentioning

confidence: 52%

“…Let us denote by \/y* the decreasing rearrangement of the collection (| a, IV1}, \ i < k, j G N}. The distribution of (\/y¡*)¡>\ is the same as that of (l/I\),->i (cf., e.g., [9]). Hence, \YZ(y*)~x/pej\\, which is equal to 112a, Z, 11 (since (e,) is a symmetric basis), has the same distribution as ||Z,||.…”

Section: «2ki-mentioning

confidence: 99%

“…More generally, it is known (cf. [9]) that if (Yj)j>x is an i.i.d. sequence of symmetric A'-valued random variables, then the variable S -^f-yTf^'Yj is /»-stable, and we have V| e X* Eexpi(É,S)= exp(-E| (£, Yx )f/{Cp)p).…”

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confidence: 99%

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On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

Pisier¹

1983

Trans. Amer. Math. Soc.

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“…We can now state the representation which we will use. This result follows from [9] (for more details, see [10]). More generally, it is known (cf.…”

mentioning

confidence: 52%

Section: «2ki-mentioning

confidence: 99%

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On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

Pisier¹

1983

Trans. Amer. Math. Soc.

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“…If p îs symmetric p-stable with the spectral measure a then by (4) and (7) (2) For symmetric p-stable random variable on R with characteristic function exp(-|i*|p) we have Pi\X\ < t) < ^_1T(1 + 1/p)/ because T(l + l/p)/7r is precisely the value of the density at zero, hence the constant ApBPir) must tend to infinity at least like T(l + 1/p) when p -» 0.…”

Section: Estimation Of Constantsmentioning

confidence: 99%

“…The purpose of this note is both to extend ( 1 ) to the case of symmetric Testable measures, 0 < p < 2, and to give another proof for (1). Our method consists in using a series representation of stable random vectors obtained in [4].…”

Section: Introduction and Preliminary Factsmentioning

confidence: 99%

Stable Measure of a Small Ball

Lewandowski¹,

Ryznar²,

Zak³

1992

Proceedings of the American Mathematical Society

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Variations in multiscale curvature distribution and signatures of LiDAR DTM errors

Sofia

Pirotti

Tarolli

2013

Earth Surf Processes Landf

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The development of high resolution LiDAR digital terrain models (DTMs) has enabled the exploration of the statistical signature of morphology on curvature distributions. This work analyzes Minimum Curvature distributions to identify the statistical signature of two types of LiDAR-DTM errors (outliers and striping artifacts) in the derived estimates, rather than morphology itself. The analysis shows the importance of modeling these errors correctly, in relation to the scale of analysis and DTM resolution, in order to have reliable curvature estimates. Nine DTMs of different morphological areas are considered, and grouped into a training dataset (without errors) and a test dataset (with errors). In the training dataset, the original DTMs are considered as true values; errors are then applied to these data. Minimum Curvature is computed at multiple scales from each DTM: changes in curvature distributions due only to morphology and scale are characterized from the original data; error effects are then identified from the datasets with simulated errors, and validated against the test dataset. The analysis shows that outliers and striping artifacts can be realistically simulated by heavily left tailed distributions. For DTMs without errors, the scale-dependent change in curvature distribution is primarily controlled by real morphology. When DTMs include errors, curvature distributions become controlled by these errors, whose propagation depends on error distribution, error spatial correlation, and the scale of analysis. This study shows that the curvature distributions are impacted upon differently by striping artifacts and outliers, and that these are clearly distinguishable from the signal of morphological features: a scale-dependent change in curvature distribution can therefore be interpreted as the signature of these specific errors, rather than morphology

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Convergence to a Stable Distribution Via Order Statistics

Cited by 173 publications

References 5 publications

On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

Stable Measure of a Small Ball

Variations in multiscale curvature distribution and signatures of LiDAR DTM errors

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