1. Stochastic models - time series.
Random process.
an infinite collection of consistent distributions
probabilities exist
Random function.
family of random variables, {Y(t;), t  Z, }
Z = {0,±1,±2,...},  a sample space

2. Specified if given
F(y1,...,yn;t1 ,...,tn ) = Prob{Y(t1)y1,...,Y(tn )yn }
n = 1,2,... E.g. t in Z or R
F's are symmetric in the sense
F(y;t) = F(y;t),  a permutation
F's are compatible
F(y1 ,...,ym ,,...,;t1,...,tm,tm+1,...,tn} = F(y1,...,ym;t1,...,tm)
m+1  n = 2,3,...

3. Finite dimensional distributions
First-order
F(y;t) = Prob{Y(t)  t}
Second-order
F(y1,y2;t1,t2) = Prob{Y(t1)  y1 and Y(t2)  y2}
and so on

4. There may be densities
joint density
Prob{x < Y(t+u) < x+dx and y < Y(t) < y+ dy}
= f(x,y,u) dxdy

5. Extensions.
The Y(t) may be discrete, angles, proportions, vectors, ...
Kolmogorov extension theorem. To specify a stochastic process give the distribution of any finite subset {Y(1),...,Y(n)} in a consistent way,  in some A

6. The algebra.
(*)
Extend to case of > 2 variables

7. Normal process/series.
Finite dimension distributions multivariate normal
Multivatiate normal.
Entries linear combinations of i.i.d standard normals
Y =  +  Z
: s by 1 : s by r Y: s by 1 Z: Nr(0,I) I: r by r identity
E(Y) =  var(Y) = ' s by s
Y = [Y1τ Y2τ]τ
Conditional marginals linear in Y2 when condition on it

8. Other methods
i) Y(t;), : random variable measurability
ii) urn model
iii) probability on function space
iv) analytic formula
Y(t) =  cos(t + )
, : fixed : uniform on (-,]

9. Moment functions.
Mean function
cY(t) = E{Y(t)} =  y dF(y;t)
=  y f(y;t) dy if continuous
=  yjf(yj; t) if discrete
E{1Y1(t) + 2Y2(t)} =1c1(t) +2c2(t)
Signal plus noise model: Y(t) = S(t) + Z(t) e.g. S(.) fixed/random
E{Y(t)} = S(t) fixed case

10. Y S^ N^

11. Shumway and Stoffer (2011)

12. DRB (1975)

13. Second-moments.
autocovariance function
cYY(s,t) = cov{Y(s),Y(t)} = E{Y(s)Y(t)} - E{Y(s)}E{Y(t)}
non-negative definite
 jkcYY(tj , tk )  scalars 
var { jY(ti)}
crosscovariance function
c12(s,t) = cov{Y1(s),Y2(t)}

15. Shumway and Stoffer (2011)

16. Stationarity.
Joint distributions,
{Y(t+u1),...,Y(t+uk-1),Y(t)},
do not depend on t for k=2,3,...
Often reasonable in practice, particularly for some time stretches
Replaces "identically distributed" (i.i.d.)

17. Cryer and Chan (2008)

18. mean
E{Y(t)} = cY for t in Z
autocovariance function
cov{Y(t+u),Y(t)} = cYY(u) t,u in Z u: lag
= E{Y(t+u)Y(t)} if mean 0
autocorrelation function (u) = corr{Y(t+u),Y(t)}, |(u)|  1
crosscovariance function
cov{X(t+u),Y(t)} = cXY(u)

19. DRB

21. DRB (1975)

23. acf’s and ccf