Distribution of the height of local maxima of Gaussian random fields

Dan Cheng, Armin Schwartzman

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


Let {f(t) : t ∈ T} be a smooth Gaussian random field over a parameter space T, where T may be a subset of Euclidean space or, more generally, a Riemannian manifold. We provide a general formula for the distribution of the height of a local maximum ℙ{f(t0) > u|t0 is a local maximum of f(t)} when f is non-stationary. Moreover, we establish asymptotic approximations for the overshoot distribution of a local maximum ℙ{f(t0) > u+v|t0 is a local maximum of f(t) and f(t0) > v} as v→∞$v\to \infty $. Assuming further that f is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when T is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

Original languageEnglish (US)
Pages (from-to)213-240
Number of pages28
Issue number2
StatePublished - Dec 11 2015
Externally publishedYes


  • Euler characteristic
  • Gaussian orthogonal ensemble
  • Height
  • Isotropic field
  • Local maxima
  • Overshoot
  • Riemannian manifold
  • Sphere

ASJC Scopus subject areas

  • Statistics and Probability
  • Engineering (miscellaneous)
  • Economics, Econometrics and Finance (miscellaneous)


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