An Introduction to the Theory of Point Processes by GUJARATI

By GUJARATI

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K)}, that is independent of the order in which the random variables are written down. The second embodies an essential requirement: it must be satisfied if there is to exist a single probability space Ω on which the random variables can be jointly defined. 28 9. 22), which express the fact that the random variables so produced must fit together as measures. VI (Measure Requirements). (a) Additivity. For every pair A1 , A2 of disjoint Borel sets from BX , the distribution F3 (A1 , A2 , A1 ∪A2 ; x1 , x2 , x3 ) is concentrated on the diagonal x1 + x2 = x3 .

By C0 = {N : N (A) = 0} and Cr = {N : N (Br ) = 0 < N (Br−1 )}. r Then P (C0 ∪ C1 ∪ · · ·) = 1, and N (A) = i=1 N (Ai ) + N (Br ) on Cr . Also, on Cr , it follows from 0 = N (Br ) = limn→∞ ζn (Br ) that N (Ai ) = 0 for ∞ ∞ i ≥ r + 1 and hence i=r+1 N (Ai ) = 0 on Cr . Because P r=0 Cr = 1, it now follows that N is countably additive on R. s. to a countably additive boundedly finite nonnegative integer-valued measure on BX . This extension, with the appropriate modification on the P -null set where the extension may fail, provides the required example of a point process with avoidance function P {N (A) = 0} = ψ(A) (A ∈ R) satisfying conditions (i)–(iv).

S. , j=1 where the last equality follows by induction from the previous result. Then n var Wn = µ n j=1 Aj = µ(Aj ), j=1 20 9. Basic Theory of Random Measures and Point Processes and if W = ξ ∞ j=1 Aj , we must have var(Wn − W ) → 0. 24)]. 22). On the other hand it is not true that for almost all ω the realizations are signed measures. To see this, let {A1 , . . , An } be a finite partition of A and set n |ξ(Aj )|. s. over all possible partitions. But n E |ξ(Aj )| = E(Yn ) = j=1 and n µ(Aj ) j=1 1/2 ≥ n j=1 2 π 1/2 n µ(Aj ) , j=1 µ(Aj ) max1≤j≤n µ(Aj ) 1/2 1/2 = µ(A) max1≤j≤n µ(Aj ) 1/2 , so E(Yn ) can be made arbitrarily large by choosing a partition for which max1≤j≤n µ(Aj ) is sufficiently small.

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