1. Point processes
rates are a point process concern

2. Point process data
points along the line
radioactive emissions, nerve cell firings, …
Describe by: a) 0 1 < 2 < ... < N < T in [0,T)
b) N(t) = #{ j | 0 j < t}, a step function
c) counting measure N(I) =
d) Y0 = 1, Y1 = 2 - 1 , ..., YN-1 = N - N-1
intervals  0
e) Y(t) = j (t-j) = dN(t)/dt
(.): Dirac delta function

3. Data displays

4. Point process data can arise from crossings

5. empirical rate: N(T)/T
slope
empirical running rate: [N(t+)-N(t- )]/2 
change?

6. Stacking

7. Clustering

8. Properties of the Dirac delta, (.).
a generalized function, Schwartz distribution
(0) =  (t) = 0, t  0
density function of a r.v.,Ƭ, that = 0 with probability 1
cdf H(t) = 0, t<0 H(t) =1, t  0
for suitable g(.), E(g(Ƭ)) =
g(.): test function

9. Y(t) = j (t-j) = dN(t)/dt
= N(g)
Can treat a point process as an "ordinary" time series using 
orderly: points are isolated no twins
In survival analysis just 1 point
Might analyze interval series Yk = k+1 - k , non-negative

10. Vector-valued point process
points of several types
N(t)
Y(t) = dN(t)/dt

12. Marked point process.
{j , Mj }
mark Mj is associated with time j
examples: earthquakes, insurance
If marks real-valued: jump or cumulative process
point process if Mj = 1

13. Y(t) = j Mj (t-j ) = dJ(t)/dt

14. stacking

15. Sampled time series, hybrid. X(j )
Computing.
can replace {j} by t.s. Yk = dN(t) with k = [j/dt]
[.]: integral part
Point processes are very, very basic in science
particle vs. wave theory of light