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

Asymptotically normal.

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

Nonstationary case. p N (t)

Second-order.

Bivariate p.p.

Volkonski and Rozanov (1959); If N T (I), T=1,2,… sequence of point processes with p N T  0 as T   then, under further regularity conditions, sequence with rescaled time, N T (I/p N T ), T=1,2,…tends to a Poisson process. Perhaps I NM T (u) approximately Poisson, rate  Tp NM T (u) Take:  = L/T, L fixed N T (t) spike if M spike in (t,t+dt] and N spike in (t+u,t+u+L/T] rate ~ p NM (u)  /T  0 as T   N T (IT) approx Poisson I NM T (u) ~ N T (IT) approx Poisson, mean  Tp NM (u)

Variance stabilizing transfor for Poisson: square root

For large mean the Poisson is approx normal