1. Estimation of the spectral density function

2. The spectral density function, f(l) The spectral density function, f(x), is a symmetric function defined on the interval [-p,p] satisfying and The spectral density function, f(x), can be calculated from the autocovariance function and vice versa.

3. Some complex number results: Use

5. Expectations of Linear and Quadratic forms of a weakly stationary Time Series

6. Expectations, Variances and Covariances of Linear forms

7. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

8. Proof

9. Also since Q.E.D.

10. Theorem Let {xt:tT} be a weakly stationary time series. Let and

11. Expectations, Variances and Covariances of Linear formsSummary

12. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

13. Theorem Let {xt:tT} be a weakly stationary time series. Let and Then where and

14. Then where and Also Sr= {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

15. Expectations, Variances and Covariances of Quadratic forms

16. Theorem Let {xt:t T} be a weakly stationary time series. Let Then

17. and

19. and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0, k(h,r,s) = the fourth order cumulant = E[(xt - m)(xt+h - m)(xt+r - m)(xt+s - m)] - [s(h)s(r-s)+s(r)s(h-s)+s(s)s(h-r)] Note k(h,r,s) = 0 if {xt:t T}is Normal.

20. Theorem Let {xt:t T} be a weakly stationary time series. Let Then

22. where and

23. Examples The sample mean

24. Thus and

25. Also

26. and where

27. Thus Compare with

28. If g(•) is a continuous function then: Basic Property of the Fejer kernel: Thus

32. The sample autocovariancefunction The sample autocovariance function is defined by:

33. or if m is known where

34. or if m is known where

35. Theorem Assume m is known and the time series is normal, then: E(Cx(h))= s(h),

36. and

37. Proof Assume m is known and the the time series is normal, then: and

40. and

41. where

42. since

43. hence

44. Thus

45. and Finally

46. Where

47. Thus

48. Expectations, Variances and Covariances of Linear formsSummary

49. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

50. Theorem Let {xt:tT} be a weakly stationary time series. Let and Then where and