G. , O. ). , and (. , v 1 ) have the same distribution on G * 2. reversibility: (G, o, v 1 ) and (G, v 1 , o) have the same distribution on G * *

, ) = P{ x 1 , x 2 ? ? } in terms of the capacity functional of ?

, Let ? be a general coverage model with capacity functional T ? , generated by a set process ? f with Laplace transform L ? f . Prove the following relation for

, Prove the expressions (14.1.13) and (14.1.14) for the volume fraction and the covariance function of a homogeneous germ-grain Boolean model

, Let ? f = i ? F i be a hard core set-process on R d , (i.e. the set-atoms F i do not overlap) with F i being balls

F. P{-k-?-?-}-=-e, Argue that for any connected K the full coverage functional is equal to the mean measure of ? f on

, Hint: K ? ? iff K is entirely contained in of the spherical set-atoms F i

, Let ? be a homogeneous germ-grain Boolean coverage model driven by a homogeneous Poisson process with intensity ?, 0 < ? < ? on R d and generic grain Z 0, p.1

, Let? = X i ,Z i be a stationary germ-grain model on R d as in (14.1.8), considered on some stationary framework (see the corresponding lesson) (?, A, {? x } x?R d , P)

, Using the flow of the stationary framework this means Z i = Z ? ? X i for some random closed set Z defined on the considered probability space. Stationary Voronoi tessellation and the Johnson-Mehl coverage model are examples of such process. Any stationary germ-grain process with

, Denote by ? f the corresponding set process (14.1.9) and its set-atoms by F i := X i + Z i

, Examples of stationary, ergodic random closed sets are homogeneous Boolean germ-grain models on R d . The case of deterministic grains follows immediately form the mixing property (implying ergodicity) of the homogeneous Poisson point process. The general case of (i.i.d.) compact grains requires some additional argument; cf, 1996.

, Consider the following two subsets of the plane, vol.2

, ? Let F 1 = {(x, y) ? R 2 : y = sin(1/x), x = 0} sup{(0, y) : y ?

, ? Let F 2 = {(x, y) ? R 2 : y = 1/|x|, x = 0}. Prove that F 2 is a closed and disconnected set

, ? F 2 ? {?} is closed and connected in the one-point (Alexandroff) compactifiation R 2 ? {?} of R 2 where the extra point ? is considered to be the limit of all sequences x n on R 2

, Hint: Denote by I i the indicator of the event that the grain X i + B X i (R i ) belongs to an unbounded component. Use the Campbell-Mecke-Matthes theorem

, Prove for a one-dimensional homogeneous Boolean model that E [R] < ? implies ? c = ? and E [R] = ? implies ? c = 0. (Strictly speaking in this latter case it is no longer a Boolean model

, Computer exercise. Estimate the normalized critical intensity ? * c for planar Boolean model with fixed spherical grains. (Recall, ? * c ? 1.1281). Use R with spatstat and igraph and SGCS

, Use the command clusters to obtain the connected components of the graph. For a given realization of germs, plot the fraction of nodes in the largest and second largest cluster in function of the grain radius. Estimate the critical radius as the radius when second largest component attains its maximal value, Simulate Poisson process with unit intensity in a reasonably large window. Construct the matrix of distances between the simulated points

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