Central limit theorems for the excursion set volumes of weakly dependent random fields

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.

• Publication year

  2010

• Venue

  Bernoulli

• Publication date

  2010-05-04

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.3150/10-BEJ339arXiv 1005.0483
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• A statistical version of the central limit theorem for excursion set volumes is also considered.

  Confidence 0.84

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• A formula for the covariance matrix of the limiting distribution is provided.

  Confidence 0.94

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• Gaussian fields and shot noise fields are treated as important special cases of the general excursion-set volume limit theory.

  Confidence 0.90

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• Multivariate central limit theorems are established for excursion set volumes of stationary quasi-associated random fields on R^d.

  Confidence 0.97

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review

• covariance matrix

  matrix, statistical quantity

  The matrix collecting pairwise covariances of the components of the limiting Gaussian vector.

  Aliases: covariance matrices

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• excursion set volumes

  quantity, random field functional

  The volumes of the regions where a random field exceeds a specified level.

  Aliases: excursion volumes, excursion-set volumes

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• gaussian fields

  random field

  Random fields whose finite-dimensional distributions are Gaussian.

  Aliases: Gaussian random fields

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• limiting distribution

  distribution, statistical quantity

  The asymptotic distribution approached by the normalized excursion-set volume vector.

  Aliases: limit distribution

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• multivariate central limit theorem

  theorem, result

  A limit theorem describing the joint Gaussian asymptotic behavior of several normalized excursion-set volume variables.

  Aliases: multivariate CLT, multivariate CLT for excursion set volumes

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• shot noise fields

  random field

  Random fields constructed from superposed random impulses or kernels centered at random locations.

  Aliases: shot-noise fields

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• stationary quasi-associated random fields

  random field, probabilistic model

  Random fields on R^d that are stationary and satisfy a quasi-association dependence condition.

  Aliases: quasi-associated random fields

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review
• statistical version of the clt

  theorem, result

  A central limit theorem formulation intended for statistical use with excursion-set volume functionals.

  Aliases: statistical CLT

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewB (s683577b42) review

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