1. Point processes. Some special cases.
1. a). (Homogeneous) Poisson.
Rate 
N(t), t in R
Approaches.

3. Can use to check homogenious Poisson assumption
Examples
Brillinger, Bryant, Segundo Biological Cybernetics 22, (1976)

6. DebrisEnv.gif

10. 1. b) Inhomogeneous Poisson.

11. Time substitution. N(t) Poisson rate (t)

12. Zero probability function. Characterizes simple N
(I) = Pr{N(I;)=0}, for all bounded Baire
For Poisson, (t)
(I) =exp{-(I)} capital 
M(I) = I (t)dt
Risk analysis. Poisson events, rate , period T
Prob{event in T} = 1 - exp{-|T|}  |T|

13. 2. Doubly stochastic (stationary) Poisson.

14. 3. Self-exciting process.

15. 4. Renewal process.

16. Waiting time paradox.
Homogeneous Poisson, Y's i.i.d. exponentials
Forward recurrence time - waiting time
Backward recurrence time
Time between events
all exponential parameter 

17. 5. Cluster process.

18. Operations on point processes. To produce other p.p.'s
Superposition
M(t) + N(t)
Thinning
i Ij (t-j), Ij=0 or 1
Time substitution
N(t) = M(Q(t)) Q monotonic nondecreasing
dN(t) = dM(Q(t))q(t)dt
Random translation
i (j + uj)

19. Probability generating functional.
G[] = E{exp(log (t)) dN(t)}
For Poisson
G[] = exp{((t)-1)(t)dt}

20. Limiting cases.
Poisson
1. Superpose p.p.
2. Rare events
3. Deletion

21. Can make a p.p. into a t.s.
Y(t) =  a(t - j) - < t < 
=  a(t-u)dN(u)
for some suitable a(.)

22. (Stationary) interval functions.
Y(I), I an interval
{Y(I1+u),...,Y(IK+u)} ~ {Y(I1),...,Y(IK)}
Y(I) = I [(exp{it}-1)/i]dZY()
cov{dZY(),dZY()| = (-)fYY()d d
Cov(Y(I),Y(J)} = I J cov{dY(s),dY(t)}
e.g. m.p.p.