Download A Comprehensive Introduction to Differential Geometry Volume by Michael Spivak PDF

By Michael Spivak

Publication by way of Michael Spivak, Spivak, Michael

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1. If ν is a translation-invariant measure on Rd then ν(B) = cλd (B) for some c ≥ 0. Corollary 1 If X is a stationary point process in Rd , then its intensity measure ν is a constant multiple of Lebesgue measure λd . The constant c in Corollary 1 is often called the intensity of X. 2. Suppose the intensity measure ν of a point process X in Rd satisfies β(u) du ν(B) = B for some function β. Then we call β the intensity function of X. If it exists, the intensity function has the interpretation that in a small region dx ⊂ Rd P(N (dx) > 0) ∼ EN (dx) ∼ β(x) dx.

We have Rd 1{u, v ∈ b(x, r)} dx = Rd = Rd 1{u ∈ b(x, r)}1{v ∈ b(x, r)} dx 1{x ∈ b(u, r)}1{x ∈ b(v, r)} dx = λ2 (b(u, r) ∩ b(v, r)). Hence the second moment density of the Mat´ern cluster process is g2 (u, v) = α2 μ2 + α μ2 λ2 (b(u, r) ∩ b(v, r)). 4 Second Moments for Stationary Processes For a stationary point process in Rd , there is a ‘disintegration’ of the second moment measure. Stationarity implies E [N (A + v)N (B + v)] = E [N (A)N (B)] for all v ∈ Rd . Thus ν2 , ν[2] are invariant under simultaneous shifts (x, y) → (x + v, y + v).

3 Conditioning In the study of a point process we are often interested in properties relating to a typical point of the process. This requires the calculation of conditional probabilities of events given that there is a point of the process at a specified location. It leads to the concept of the Palm distribution of the point process, and the related Campbell-Mecke formula [42]. These tools allow us to define new characteristics of a point process, such as the nearest neighbour distance distribution function G.

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