1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What is a time series? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Time series versus iid data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Typical assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Fundamental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Ergodic property with a constant limit . . . . . . . . . . . . . . . . . . . 5
2.1.2 Strict Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Weak Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Weak stationarity and Hilbert spaces . . . . . . . . . . . . . . . . . . . . 9
2.1.5 Ergodic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.6 Sufficient conditions for the a.s. ergodic property with a constant limit. . . . . . . . . . . 26
2.1.7 Sufficient conditions for the L2-ergodic property with a constant limit . .. . . . .. . . 27
2.2 Specific assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Linear processes in L2(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Linear processes with E(X2t ) = ∞ . . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 Multivariate linear processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.6 Restrictions on the dependence structure . . . . . . . . . . . . . . . . . 49
3 Defining probability measures for time series . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Finite dimensional distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Transformations and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Conditions on the expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Conditions on the autocovariance function . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Positive semidefinite functions . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Spectral distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.3 Calculation and properties of F and f . . . . . . . . . . . . . . . . .
4 Spectral representation of univariate time series . . . . . . . . . . . . . . . . . . . 81
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Harmonic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Extension to general processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Stochastic integrals with respect to Z . . . . . . . . . . . . . . . . . . . . 84
4.3.2 Existence and definition of Z . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.3 Interpretation of the spectral representation . . . . . . . . . . . . . . 97
4.4 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.1 Relationship between ReZ and ImZ . . . . . . . . . . . . . . . . . . . . 98
4.4.2 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.3 Overtones . . . . . . . . . .
About the Author:
Jan Beran is Professor of Statistics at the Department of Mathematics and Statistics at the University of Konstanz, Germany. After completing his Ph.D. in mathematics at the ETH Zurich, Switzerland, he worked at several universities in the USA and at the University of Zurich in Switzerland. He has a broad range of interests, from long-memory processes and asymptotic theory to applications in finance, biology, and musicology.
