1. Stochastic time series.
Random process.
an infinite collection of consistent distributions
probabilities exist
Random function.
a family of random variables, e.g. {Y(t), t in Z}

2. Specified if given
F(y1,...,yn;t1 ,...,tn ) = Prob{Y(t1)y1,...,Y(tn )yn }
that are symmetric
F(y;t) = F(y;t),  a permutation
compatible
F(y1 ,...,ym ,,...,;t1,...,tm,tm+1,...,tn} = F(y1,...,ym;t1,...,tm)

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

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

6. 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)
vector-valued case
mean level - signal plus noise: S(t) + (t) S(.): fixed

7. 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 
crosscovariance function
c12(s,t) = cov{Y1(s),Y2(t)}

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

9. 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)

10. Higher order moments and cumulants.
multilinear functional
0 if some subset of variantes independent of rest
0 of order > 2 for normal
normal is determined by its moments

11. Product moment functions.
mY...Y (t1 ,...,tk ) = E{Y(t1 )...Y(tk )}
Cumulant functions.
cY...Y (t1 ,...,tk ) = cum{Y(t1 ),...,Y(tk )}