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The original (non-specific) Rice formula◮ U u ≡ U u (f , T ) ∆ = #{t ∈ T : f (t) = u, ḟ (t) > 0}◮ Basic conditions on f : Continuity and differentiability◮ Plus (boring) side conditions to be met later
The original (non-specific) Rice formula◮ U u ≡ U u (f , T ) ∆ = #{t ∈ T : f (t) = u, ḟ (t) > 0}◮ Basic conditions on f : Continuity and differentiability◮ Plus (boring) side conditions to be met later◮ Kac-Rice formula:∫ ∫ ∞E {U u } = y p t (u, y) dydtT 0∫ ∫ ∞= p t (u) y p t (y ∣ ∣u) dydt∫T0{= p t (u) E |ḟ (t)| 1 (0,∞) (ḟ (t)) ∣ }f (t) = u dtTw<strong>here</strong> p t (x, y) is the joint density of (f (t), ḟ (t)),p t (u) is the probability density of f (t), etc.
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 40 and 41: 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
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The Gaussian case: What can/can’t
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The Gaussian-related casef (t) = (f
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The Gaussian-related casef (t) = (f
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The Gaussian Kinematic Formula (GKF
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The perturbed-Gaussian case◮ A ph
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Applications I: Exceedence probabil
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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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