By S Abhyankar

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By the Poisson ∞ P Ir + Irc ∈ A | Φ x : |x| ≤ r = n P{ IΦe (0) ∈ A } = P Φ x : |x| ≤ r = n . 1 that conditioned on Φ x : |x| ≤ r = n, with n > 0, the random variable Ir can be represented as the sum of n independent random variables, distributed as L(0, x, m) where x and m have joint distribution 1 Λ Fx (dm)Λ(dx) . 17) P Ir + Irc ∈ A Φ x : |x| ≤ r = n = 0. Consequently, P{ IΦe (0) ∈ A } ≤ P Φ x : |x| ≤ r = 0 → 0 when r → ∞ because Λ(Rd ) = ∞. This completes the proof. 3 Interference Field as Shot-Noise Consider a collection of transmitters distributed in the space and sharing a common radio medium.

For a given x ∈ Rd and φ ∈ M, define the distance R∗ (x) = R∗ (x, φ) = minxi ∈φ |xi − x| from x to its nearest neighbor in φ ∈ M. Note that the min is well defined due to the fact that φ is locally finite, even if arg minxi ∈φ |xi − x| is not unique. p. with intensity measure Λ and let Px! be its reduced Palm measure given a point at x. By Slivnyak’s theorem inria-00403039, version 4 - 4 Dec 2009 Px! ({φ : R∗ (x, φ) > r }) = P{ Φ(Bx (r)) = 0 } = e−Λ(Bx (r)) , where Bx (r) is the (closed) ball centered at x and of radius r.

A random compact set S(Φ) is called a stopping set if one can say whether the event { S(Φ) ⊂ K } holds or not knowing only the points of Φ in K. Remark: It can be shown that if S = S(Φ) is a stopping set, then for all φ ∈ M, inria-00403039, version 4 - 4 Dec 2009 S(Φ) = S(Φ ∩ S(Φ) ∪ φ ∩ S c (Φ)) where S c is the complement of S. In other words, all modifications of Φ outside the set S(Φ) have no effect on S(Φ). Here is a very typical example of a stopping set. 2 (k th smallest random ball). , Rk∗ = Rk∗ (Φ) = min{r ≥ 0 : Φ(B0 (r)) = k}.