An Introduction to the Theory of Point Processes by GUJARATI PDF

By GUJARATI

ISBN-10: 0072427922

ISBN-13: 9780072427929

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Example text

S. countably additive. This is enough to show that the sequence {yi } induces a counting measure on X , and so sets up a mapping from its probability space into NX# . To show that N (·) is a point process, the critical step is to show that this mapping is measurable. VIII it is enough to show that for each Borel set A, N (A) is a random variable. To this end, for each k = 1, 2, . . and ω ∈ / E0 , write Nk (A, ω) = k i=1 δyi (ω) (A). 16) 14 9. Basic Theory of Random Measures and Point Processes Because yi is an X -valued random variable, and A is a Borel subset of X , each δyi (A) is a random variable for i = 1, .

K), N (B) = 0}. 13) Proof. For k = 1 we have P{N (A1 ) > 0, N (B) = 0} = P0 (B) − P0 (A1 ∪ B) = ∆(A1 )P0 (B). 5). 13) with B the null set, ∆(A1 , . . , Ak )P0 (∅) = P{N (Ai ) > 0 (i = 1, . . , k)}. XV in a characterization of the avoidance function. 13) can be expressed immediately in terms of the avoidance function. 12) is that it leads to a succinct description of the fidi distributions of a point process when, for suitable sets Ai , N (Ai ) is ‘small’ in the sense of having P{N (Ai ) = 0 or 1 (all i)} ≈ 1.

Basic Theory of Random Measures and Point Processes a unique and consistent family of fidi distributions? This is the underlying reason for the importance of the studies by Moyal (1962) and Harris (1963), who were the first to set up a systematic theory of point processes in these terms. It also opens up the way for extensions to point processes on general types of spaces. On the other hand, the fidi distributions do not always provide the most convenient framework for examining the structure of particular models.

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An Introduction to the Theory of Point Processes by GUJARATI


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