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Example: The Brownian sheet on R NE{W (t)} = 0E{W (s)W (t)} = (s 1 ∧ t 1 ) × · · · × (s N ∧ t N ).w<strong>here</strong> t = (t 1 , . . . , t N ).◮ Replace j by j = (j 1 , . . . , j N )⇒ϕ j (t) = 2 N/2 N ∏i=12(2j i + 1)π sin ( 12 (2j i + 1) πt i)W (t) = ∑ · · · ∑ξ j1 ,...,j Nϕ j1 ,...,j N(t)j 1◮ N = 1W is standard Brownian motion.The corresponding expansion is due to Lévy, and thecorresponding RKHS is known as Cameron-Martin space.j N
Constant variance Gaussian processes◮ We know that∑◮ f (t) = ξj ϕ j (t)◮ ḟ (t) = ∑ ξ j ˙ϕ j (t)∑◮ E{ḟ (s)ḟ (t)} = ˙ϕ j (s) ˙ϕ j (t)◮ E{f (s)ḟ (t)} = ∑ ϕ j (s) ˙ϕ j (t)
Page 4 and 5: Our heroesMarc KacStephen O. Rice19
Page 6 and 7: Real roots of algebraic equations (
Page 10 and 11: Zeroes of complex polynomialsf (z)
Page 12 and 13: Zeroes of complex polynomialsf (z)
Page 14 and 15: Thinking more generally:1: f : R N
Page 16 and 17: Thinking more generally:1: f : R N
Page 18 and 19: The original (non-specific) Rice fo
Page 30 and 31: Constructing Gaussian Processes◮
Page 32 and 33: Constructing Gaussian Processes◮
Page 34 and 35: Existence of Gaussian processesTheo
Page 36 and 37: Existence of Gaussian processesTheo
Page 38 and 39: Example: The Brownian sheet on R NE
Page 42 and 43: Constant variance Gaussian processe
Page 50 and 51: Gaussian Kac-Rice with no simplific
Page 52 and 53: Gaussian Kac-Rice with no simplific
Page 54 and 55: Some pertinent thoughts◮ Real roo
Page 56 and 57: Some pertinent thoughts◮ Real roo
Page 58 and 59: The Kac-Rice “Metatheorem”
Page 60 and 61: The Kac-Rice “Metatheorem”◮ T
Page 62 and 63: The Kac-Rice “Metatheorem”◮ T
Page 68 and 69: The Kac-Rice Conditions (the fine p
Page 70 and 71: Higher (factorial) moments◮ Facto
Page 72 and 73: The Gaussian case: What can/can’t
Page 80 and 81: The Gaussian-related casef (t) = (f
Page 82 and 83: The Gaussian-related casef (t) = (f
Page 84 and 85: The Gaussian Kinematic Formula (GKF
Page 86 and 87: The perturbed-Gaussian case◮ A ph
Page 88 and 89: Applications I: Exceedence probabil
Page 90 and 91:
Applications I: Exceedence probabil
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Applications II: Local maxima on th
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Applications III: Local maxima on M
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Applications III: Local maxima on M
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Applications IV: Longuet-Higgins an
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Applications V: Higher moments and
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Applications VI: Poisson limits and
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Applications VI: Poisson limits and
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Applications VII: Eigenvalues of ra
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Applications VII: Eigenvalues of ra
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Appendix I: The canonical Gaussian
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Exceedence probabilities for canoni
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◮ We needP { sup〈U, t〉 ≥ u/
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Appendix II: Stationary and isotrop
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Appendix II: Stationary and isotrop
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◮ Elementary considerations give{
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◮ Elementary considerations give{
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Appendix III: Regularity of Gaussia
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Appendix III: Regularity of Gaussia
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Special cases of the entropy result
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Special cases of the entropy result
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Appendix IV: Borell-Tsirelson inequ
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Appendix IV: Borell-Tsirelson inequ
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