# An Introduction to the Theory of Point Processes by GUJARATI

By GUJARATI

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Extra resources for An Introduction to the Theory of Point Processes

Sample text

Let the measure ηk on X × K, with X = R and K = Z+ , have unit atoms at all the points {(i + j/2k , 2k ): i = 1, 2, . . ; j = 0, 1, . . , 2k − 1} so ηk has boundedly ﬁnite support in R × Z+ , and let Nr = rk=1 ηk . Show that each Nr is an element of NX#∗ ×K but that their limit is not. 3 Show that an MPP can be simple even if its ground process is not simple. 4 Let N be a simple point process on BX ×K , and K a ﬁxed bounded Borel set in K. Show that NK (A) = N (A × K) (bounded A ∈ BX ) deﬁnes a simple point process.

Ak }, z Fk (A1 , A2 , A3 , . . , Ak ; dx1 , z − x1 , x3 , . . 7) = Fk−1 (A1 ∪ A2 , A3 , . . , Ak ; z, x3 , . . , xk ). Proof. VI(a) and therefore necessary. We show that it is also suﬃcient. Let us ﬁrst point out how the extension from disjoint to arbitrary families of sets can be made. Let {B1 , . . , Bn } be any such arbitrary family. Then there exists a minimal family {A1 , . . , Ak } of disjoint sets (formed from the nonempty intersections of the Bi and Bic ) such that each Bi can be represented as a ﬁnite union of some of the Aj .

Proof. X. 17) when written in the form r P{ζn (A) = k} = lim P{N (A) ≤ r} = 1 lim lim r→∞ n→∞ k=0 r→∞ and expresses the fact that a point process N is boundedly ﬁnite. For the suﬃciency, it is clear from (i) and (ii) that we can construct an indicator process Z on bounded A ∈ R with ﬁdi distributions (for any ﬁnite number k of disjoint bounded A1 , . . 18a) ⎫ Pr{Z (Ai ) = 1 (i = 1, . . , k} = ∆(A1 , . . 18b) = ∆(A1 , . . , Aj−1 , Aj+1 , . . Ak ) ψ(Aj ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ k Pr{Z (Ai ) = 0 (all i)} = ψ i=1 Ai ); nonnegativity is ensured by (i), summation to unity by (ii), and marginal consistency reduces to ∆(A1 , .