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Probability generating functionals

The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined analogously with respect to any non-negative bounded function v {\displaystyle \textstyle v} on R d {\displaystyle \textstyle \mathbb {R} ^{d}} such that 0 ≤ v (x) ≤ 1 {\displaystyle \textstyle 0\leq v(x)\leq 1}. For a point process N {\displaystyle \textstyle {N}} the probability generating functional is defined as:

where the product is performed for all the points in N {\textstyle N}. If the intensity measure Λ {\displaystyle \textstyle \Lambda } of N {\displaystyle \textstyle {N}} is locally finite, then the G {\textstyle G} is well-defined for any measurable function u {\displaystyle \textstyle u} on R d {\displaystyle \textstyle \mathbb {R} ^{d}}. For a Poisson point process with intensity measure Λ {\displaystyle \textstyle \Lambda } the generating functional is given by:

which in the homogeneous case reduces to