1. The Empirical FT continued. What is the large sample distribution of the EFT?

2. The complex normal.

3. Theorem. Suppose p.p. N is stationary mixing, then

4. Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments Proof. Write

5. Consider

6. Comments. Already used to study rate estimate Tapering makes Get asymp independence for different frequencies The frequencies 2  r/T are special, e.g.  T (2  r/T)=0, r  0 Also get asymp independence if consider separate stretches p-vector version involves p by p spectral density matrix f NN ( )

7. Estimation of the (power) spectrum. An estimate whose limit is a random variable

8. Some moments. The estimate is asymptotically unbiased Final term drops out if = 2  r/T  0 Best to correct for mean, work with

9. Periodogram values are asymptotically independent since d T values are - independent exponentials Use to form estimates

12. Approximate marginal confidence intervals

13. More on choice of L

14. Approximation to bias

15. Indirect estimate Could approximate p.p. by 0-1 time series and use t.s. estimate choice of cells?

16. Estimation of finite dimensional . approximate likelihood (assuming I T values independent exponentials)

17. Bivariate case.

18. Crossperiodogram. Smoothed periodogram.

19. Complex Wishart

20. Predicting N via M

21. Plug in estimates.

22. Large sample distributions. var log|A T |  [|R| -2 -1]/L var argA T  [|R| -2 -1]/L

23. Networks. partial spectra - trivariate (M,N,O) Remove linear time invariant effect of M from N and O and examine coherency of residuals,

26. Point process system. An operation carrying a point process, M, into another, N N = S[M] S = {input,mapping, output} Pr{dN(t)=1|M} = {  +  a(t-u)dM(u)}dt System identification: determining the charcteristics of a system from inputs and corresponding outputs {M(t),N(t);0  t<T} Like regression vs. bivariate

28. Realizable case. a(u) = 0 u<0 A( ) is of special form e.g. N(t) = M(t-  ) arg A( ) = - 

29. Advantages of frequency domain approach. many stationary processes look the same approximate i.i.d sample values assessing models (character of departure)...

30. Muscle spindle example

31. Published by AAAS R. Fuentes et al., Science 323, 1578 -1582 (2009) Fig. 2. DCS restores locomotion and desynchronizes corticostriatal activity