Inference. Estimates stationary p.p. {N(t)}, rate pN , observed for 0<t<T First-order.

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Presentation transcript:

Inference. Estimates stationary p.p. {N(t)}, rate pN , observed for 0<t<T First-order.

Asymptotically normal.

Theorem. Suppose cumulant spectra bounded, then N(T) is asymptotically N(TpN , 2Tf2 (0)). Proof. The normal is determined by its moments

Second-order.

Bivariate p.p.

Volkonski and Rozanov (1959); If NT(I), T=1,2,… sequence of point processes with pNT  0 as T   then, under further regularity conditions, sequence with rescaled time, NT(I/pNT ), T=1,2,…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,t+dt] and N spike in (t+u,t+u+L/T] rate ~ pNM(u) /T  0 as T   NT(IT) approx Poisson INMT(u) ~ N T(IT) approx Poisson, mean TpNM(u)

Variance stabilizing transfor for Poisson: square root

For large mean the Poisson is approx normal

Nonstationary case. pN(t)

A Poisson case. Rate (t) = exp{ +  cos(t + )}  = (,,,) log-likelihood l() =  log ( i) -  (t) dt 0 < t < T