PERIODICALLY CORRELATED WIDE-SENSE MARKOV PROCESSES

Table of Contents

   Content                                                                                                           Page

Part I …………………………………………………………………………….. 1

1. Introduction and Preliminary …..………..….……….………..…..………. 2

   • Aim …………….………..….……………… …………………..……… 3

   • Preliminary ..………………………………….……………………..……4

     • Probability Space …….……….………………….…….…..……. 4

     • Random Vectors ……….………………. ……….………..……..  4

     • Hilbert Space and Second order Processes ….…………………….5

     • Stationary Processes …………………….…………….….…….. 5

     • ARMA Processes ……………………………..………..………. 6

     • Orthogonality and Projection Theorem ……..…..……………….. 7

     • Periodically Correlated Processes (PC) ……………….…………7

     • Markov Processes …………………..….…………………..……8

     • PCWM Processes ……………………………….………………10

     • PAR Processes ………………………………..……..…………10

   • Literature Review…………………………………..…………………..11

   • Summary of Results ……………..….……………….………………..15

2. Univariate PCWM Processes ………………….……….………..…..………16

2.1. Introduction ……………………………..………………….……….. …17

• Covariance Structure …………………………………..….……………18

• Autoregressive Characterization….……………..…………..……..……27

3. Multivariate WM Processes ……..……………………….………………. 30

3.1. Introduction ………………………………..……….…….……………..31

3.2. Covariance Characterization……………..…….………….….……….. 31

3.3. Multivariate Stationary WM Processes…….………………..……….…38

 Content                                                                                                              Page

4. Multivariate PCWM Processes………………………….……….…………41

• Introduction ………………………………..………..………………….42

4.2. Covariance Structure …………………………………..…………..……42

4.3. Autoregressive Characterization ……………….….……..……………..46

Part II …………………………………………………………………………………………………..50

On the estimation of missing values in AR(1) model with exponential innovations…50

1. Introduction ………………..……………………………………..…………..51

2. Estimating a missing value in AR(1) model with exponential innovations…..52

3. Comparison with respect to Pitman’s measure of closeness……………….….54

4. Estimation of a missing value when the parameters are unknown…..……….59

References ……………………………………………………..………………. 62

Table of Figures

     Figure                                                                                                            Page

Figure 1. The plot of the function  for  …………………….…..58

Figure 2. The Plots of the estimated Pitman’s measure of closeness

                    (PMC),   …………………………………………………….. 60