Some Concepts of Negative Dependence

Block, Henry W.; Savits, Thomas H.; Shaked, Moshe

doi:10.1214/aop/1176993784

Cited by 261 publications

(113 citation statements)

References 10 publications

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“…Proof First we remark that a density being log-concave is equivalent to the density being a Polya frequency function of order 2 (or PF 2 , using the terminology from [4] …”

Section: Lemma 39mentioning

confidence: 99%

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Delocalising the parabolic Anderson model through partial duplication of the potential

Muirhead

Pymar

Sidorova

2017

Probab. Theory Relat. Fields

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The parabolic Anderson model on Z d with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.

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“…Proof First we remark that a density being log-concave is equivalent to the density being a Polya frequency function of order 2 (or PF 2 , using the terminology from [4] …”

Section: Lemma 39mentioning

confidence: 99%

“…Due to our analysis in Sect. 4, we know that it is sufficient to study only the contribution from paths which visit the sites in N t at most once.…”

Section: Completion Of the Proof Of Theorem 12mentioning

confidence: 99%

Delocalising the parabolic Anderson model through partial duplication of the potential

Muirhead

Pymar

Sidorova

2017

Probab. Theory Relat. Fields

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“…In another paper of Lee [18] a parallel development of negative dependence notions is given. Other studies of positive and negative dependence concepts, that start from [ 19] and extend it in various directions, can be found in the papers by Block, Savits, and Shaked [3,4].…”

Section: Journal Of Hydrologic Engineeringmentioning

confidence: 99%

The Impact of Lehmann’s Work in Applied Probability

Piegorsch

Shaked

2011

Selected Works of E. L. Lehmann

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Lehmann's works in applied probability crossed a broad spectrum of topics, and many of his papers had substantial impact. We comment on two such examples below. The first [19] provided a valuable beginning for the study and application of notions of positive and negative dependence. The second [20] also touched on issues of probability theory and the nature of random variables-in particular, on the distributional aspects of the reciprocal of a random variable. The contributions each represented are a study in contrasts, however: the former was a substantial work from which many wide-ranging contributions developed, while the second was a shorter, later-appearing piece whose impact(s) are still evolving.

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“…We first recall the definition of negative association introduced by Block et al 1 and studied in Ref. 2.…”

Section: Introductionmentioning

confidence: 99%

On the complete convergence of weighted sums for an array of rowwise negatively superadditive dependent random variables

Xinghui¹,

Li²,

Hu³

2016

ScienceAsia

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ABSTRACT:In this paper, the complete convergence and the complete moment convergence of weighted sums for an array of negatively superadditive dependent random variables are established. The results generalize the Baum-Katz theorem on negatively superadditive dependent random variables. In particular, the Marcinkiewicz-Zygmund type strong law of large numbers of weights sums for sequences of negatively superadditive dependent random variables is obtained. KEYWORDS:Baum-Katz type theorem, Marcinkiewicz-Zygmund type strong law of large numbers MSC2010: 60B10 60F15

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Some Concepts of Negative Dependence

Cited by 261 publications

References 10 publications

Delocalising the parabolic Anderson model through partial duplication of the potential

Delocalising the parabolic Anderson model through partial duplication of the potential

The Impact of Lehmann’s Work in Applied Probability

On the complete convergence of weighted sums for an array of rowwise negatively superadditive dependent random variables

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