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,...
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  t = 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. 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
Conditional marginals linear in Y2 when condition on it

5. 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 (-,]

6. There may be densities
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 A