Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes

Marcus, Michael B.; Pisier, Gilles

doi:10.1007/bf02392199

Cited by 141 publications

(89 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The proof of Theorem 1.1 follows from the following probabilistic lemma, which is implicit in [19]. We believe that this result is of independent interest.…”

Section: Results and Techniquesmentioning

confidence: 85%

See 1 more Smart Citation

Metric Structures in L 1: Dimension, Snowflakes, and Average Distortion

Lee

Mendel

2004

LATIN 2004: Theoretical Informatics

View full text Add to dashboard Cite

We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O( √ log n), yielding the first evidence that the conjectured worst-case bound of O( √ log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open in [4] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [3,16] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space. IntroductionThis paper is devoted to the analysis of metric properties of finite subsets of L 1 . Such metrics occur in many important algorithmic contexts, and their analysis is key to progress on some fundamental problems. For instance, an O(log n)-approximate max-flow/min-cut theorem proved elusive for many years until, in [18,2], it was shown to follow from a theorem of Bourgain stating that every metric on n points embeds into L 1 with distortion O(log n).The importance of L 1 metrics has given rise to many problems and conjectures that have attracted a lot of attention in recent years. to Four basic problems of this type are as follows . (We recall that a squared-ℓ 2 metric is a space (X, d) for which (X, d 1/2 ) embeds isometrically in a Hilbert space.) Each of these questions has been asked many times before; we refer to [21,22,17,11], in particular. Despite an immense amount of interest and effort, the metric properties of L 1 have proved quite elusive; *

show abstract

“…The proof of Theorem 1.1 follows from the following probabilistic lemma, which is implicit in [19]. We believe that this result is of independent interest.…”

Section: Results and Techniquesmentioning

confidence: 85%

“…The heart of our argument is the following lemma which is implicit in [19], and which seems to be of independent interest. Lemma 2.1.…”

Section: Average Distortion Euclidean Embedding Of Subsets Of Lmentioning

confidence: 99%

Metric Structures in L 1: Dimension, Snowflakes, and Average Distortion

Lee

Mendel

2004

LATIN 2004: Theoretical Informatics

View full text Add to dashboard Cite

show abstract

“…The first part of Proposition 1.1 is a consequence of Theorem 3.3 in [7], which is applied with p = 1 to the r.v's Zi -<p(Xi); see also [9] for this inequality. The second part of Proposition 1.1 is new and is proved in Section 3.…”

Section: )mentioning

confidence: 99%

“…Throughout this paper, we denote (X n ) = (Xi,... ,X n ). Let ip: R + -* R + be a continuous increasing unbounded function such that <p(0) = 0, and let Whenever <p(t) = t p ,p> 1, and when dealing with infinite sequences, it is well known that || • || V) oo is equivalent to a norm on the space l P t 0 0 of all sequences (x n , n ^ 1) of real and complex numbers such that ( The main tool used in this paper is an extension of an inequality due to Marcus and Pisier [7]. We state it in the following …”

Section: Introduction and Notationmentioning

confidence: 99%

“…Наш метод основан на простом распространении одно го неравенства из работы [7] (М. Marcus, G. Pisier) на «хвостовые» распределения слабых / р -норм независимых последовательностей ве щественных случайных величин. Наши результаты связаны также с некоторыми обобщениями закона больших чисел Марцинкевича-Зигмунда.…”

unclassified

See 1 more Smart Citation

Strong laws for sums of extreme values

Broniatowski¹,

Weber²

1997

Teor. Veroyatnost. i Primenen.

View full text Add to dashboard Cite

В статье представлен новый подход к усиленным законам боль ших чисел для сумм экстремальных значений последовательностей независимых, но не обязательно одинаково распределенных случай ных величин. Наш метод основан на простом распространении одно го неравенства из работы [7] (М. Marcus, G. Pisier) на «хвостовые» распределения слабых / р -норм независимых последовательностей ве щественных случайных величин. Наши результаты связаны также с некоторыми обобщениями закона больших чисел Марцинкевича-Зигмунда.Ключевые слова и фразы: порядковые статистики, экстремаль ные значения, законы больших чисел, закон больших чисел Марцин-кевича-Зигмунда, банаховы пространства.

show abstract

Long Range Dependence

Samorodnitsky¹

2014

Wiley StatsRef: Statistics Reference Online

View full text Add to dashboard Cite

show abstract

Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes

Cited by 141 publications

References 21 publications

Metric Structures in L 1: Dimension, Snowflakes, and Average Distortion

Metric Structures in L 1: Dimension, Snowflakes, and Average Distortion

Strong laws for sums of extreme values

Long Range Dependence

Contact Info

Product

Resources

About