Sevan methodologies revisited: Random line processes
The paper studies random line processes Z that are translation invariant in probability distribution, and whose first and second order moment measures possess continuous densities. The purpose is to review the analytical apparatus based on the concept of horizontal or vertical windows and corresponding Palm-type probability distributions that are now proved below to exist. That apparatus enables to study the relation between the quantities p k (l, α) and π k (l, α), where p k (l, α) = probability to have k hits by the lines from Z on a “test segment” of length l and direction α, while π k (l, α) = conditional probability of the same event, the condition being that the test segment lies on one of the lines from Z. Palm equations for horizontal windows have been known since long, but for vertical windows they were first put down in the last chapter of the book , under stronger condition of Euclideanmotions invariance of Z. The paper considers two different models that imply Poissonity of the probabilities p k (l, α). Translation invariant line processes can be viewed as stationary states of random dynamical arrays of countably many particles each moving with constant speed along the test line, and these models are of special interest in that context. In a model-free setting, the paper presents a formula for calculation of the conditional intensity Λ(α) = l −1Σ k kπ k (l, α). That formula includes quantities depending on the distribution of the typical vertex shape. “Sevan metodologies” have been the topic of authors plenary report at the Rasht (Iran) meeting in 2011. This usage is motivated in a special historical section below; another section is devoted to detailed description of Sevan methodologies themselves.
KeywordsCombinatorial integral geometry stochastic geometry random lines process Palm-type distribution
MSC2010 numbers60D05 60G55 52A22
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