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Statistics
Stochastic Processes
1. Foundations of Probability Theory
2. Introduction to Stochastic Processes
3. Discrete-Time Markov Chains
4. Poisson Processes
5. Continuous-Time Markov Chains
6. Renewal Theory
7. Martingales
8. Brownian Motion
9. Stochastic Calculus
10. Stationary Processes
11. Applications in Queueing Theory
12. Applications in Finance
13. Applications in Biology and Population Dynamics
14. Applications in Physics and Engineering
10.
Stationary Processes
10.1.
Stationarity Concepts
10.1.1.
Strict-Sense Stationarity
10.1.2.
Wide-Sense Stationarity
10.1.3.
Covariance Stationarity
10.1.4.
Relationships Between Concepts
10.2.
Second-Order Properties
10.2.1.
Autocovariance Function
10.2.1.1.
Definition and Properties
10.2.1.2.
Positive Definiteness
10.2.2.
Autocorrelation Function
10.2.3.
Cross-Covariance Functions
10.2.4.
Spectral Representation
10.3.
Spectral Analysis
10.3.1.
Power Spectral Density
10.3.2.
Wiener-Khinchin Theorem
10.3.3.
Spectral Distribution Function
10.3.4.
Filtering in Frequency Domain
10.4.
Ergodicity
10.4.1.
Mean Ergodicity
10.4.2.
Covariance Ergodicity
10.4.3.
Ergodic Theorems
10.4.4.
Time vs Ensemble Averages
10.5.
Gaussian Processes
10.5.1.
Definition and Properties
10.5.2.
Stationarity of Gaussian Processes
10.5.3.
Spectral Representation
10.5.4.
Karhunen-Loève Expansion
10.6.
Linear Processes
10.6.1.
Moving Average Processes
10.6.2.
Autoregressive Processes
10.6.3.
ARMA Processes
10.6.4.
Wold Decomposition
10.7.
Prediction Theory
10.7.1.
Linear Prediction
10.7.2.
Wiener Filtering
10.7.3.
Kalman Filtering
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9. Stochastic Calculus
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11. Applications in Queueing Theory