Inference. Estimates

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Inference. Estimates stationary p.p.
N(t), rate pN , observed for 0lttltT First-order.
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Asymptotically normal.
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Theorem. Suppose cumulant spectra bounded, then
N(T) is asymptotically N(TpN , 2?Tf2 (0)). Proof.
The normal is determined by its moments
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Nonstationary case. pN(t)
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Second-order.
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Bivariate p.p.
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Volkonski and Rozanov (1959) If NT(I), T1,2,
sequence of point processes with pNT ? 0 as T ?
? then, under further regularity conditions,
sequence with rescaled time, NT(I/pNT ),
T1,2,tends to a Poisson process. Perhaps
INMT(u) approximately Poisson, rate
?TpNMT(u) Take ? L/T, L fixed NT(t) spike
if M spike in (t,tdt and N spike in
(tu,tuL/T rate pNM(u) ?/T ? 0 as T
? ? NT(IT) approx Poisson INMT(u) N T(IT)
approx Poisson, mean ?TpNM(u)
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Variance stabilizing transfor for Poisson square
root
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For large mean the Poisson is approx normal