1. Point processes on the line. Nerve firing.

2. Stochastic point process. Building blocks
Process on R {N(t)}, t in R, with consistent set of distributions
Pr{N(I1)=k1 ,..., N(In)=kn } k1 ,...,kn integers  0
I's Borel sets of R.
Consistentency example. If I1 , I2 disjoint
Pr{N(I1)= k1 , N(I2)=k2 , N(I1 or I2)=k3 }
=1 if k1 + k2 =k3
= 0 otherwise
Guttorp book, Chapter 5

3. Points: ... -1  0  1  ...
discontinuities of {N}
N(t) = #{0 < j  t}
Simple: j  k if j  k
points are isolated
dN(t) = 0 or 1
Surprise. A simple point process is determined by its void probabilities
Pr{N(I) = 0} I compact

4. Conditional intensity. Simple case
History Ht = {j  t}
Pr{dN(t)=1 | Ht } = (t:)dt  r.v.
Has all the information
Probability points in [0,T) are t1 ,...,tN
Pr{dN(t1)=1,..., dN(tN)=1} =
(t1)...(tN)exp{- (t)dt}dt1 ... dtN
[1-(h)h][1-(2h)h] ... (t1)(t2) ...

5. Parameters. Suppose points are isolated
dN(t) = if point in (t,t+dt]
= otherwise
1. (Mean) rate/intensity.
E{dN(t)} = pN(t)dt
= Pr{dN(t) = 1}
j g(j) =  g(s)dN(s)
E{j g(j)} =  g(s)pN(s)ds
Trend: pN(t) = exp{+t} Cycle:  + cos(t+)  0

6. Product density of order 2.
Pr{dN(s)=1 and dN(t)=1}
= E{dN(s)dN(t)}
= [(s-t)pN(t) + pNN (s,t)]dsdt
Factorial moment

7. Autointensity.
Pr{dN(t)=1|dN(s)=1}
= (pNN (s,t)/pN (s))dt s  t
= hNN(s,t)dt
= pN (t)dt if increments uncorrelated

8. Covariance density/cumulant density of order 2.
cov{dN(s),dN(t)} = qNN(s,t)dsdt st
= [(s-t)pN(s)+qNN(s,t)]dsdt generally
qNN(s,t) = pNN(s,t) - pN(s) pN(t) st

9. Identities.
1. j,k g(j ,k ) =  g(s,t)dN(s)dN(t)
Expected value.
E{ g(s,t)dN(s)dN(t)}
=  g(s,t)[(s-t)pN(t)+pNN (s,t)]dsdt
=  g(t,t)pN(t)dt +  g(s,t)pNN(s,t)dsdt

10. 2. cov{ g(j ),  h(k )}
= cov{ g(s)dN(s),  h(t)dN(t)}
=  g(s) h(t)[(s-t)pN(s)+qNN(s,t)]dsdt
=  g(t)h(t)pN(t)dt +  g(s)h(t)qNN(s,t)dsdt

11. Product density of order k.
t1,...,tk all distinct
Prob{dN(t1)=1,...,dN(tk)=1}
=E{dN(t1)...dN(tk)}
= pN...N (t1,...,tk)dt1 ...dtk

13. Proof of Central Limit Theorem via cumulants in i.i.d. case.
Normal distribution facts.
1. Determined by its moments
2. Cumulants of order  2 identically 0
Y1, Y2, ... i.i.d. mean 0, variance 2, all moments, E{Yk}
k=1,2,3,4,... existing
Sn = Y1 + Y Yn E{Sn } = var{ Sn} = n 2
cumr Sn = n r cumr Y = cum{Y,...,Y}
cumr {Sn / n} = n r / nr/2
 0 for r = 3, 4, ...
 2 r = as n  

14. Cumulant density of order k.
t1,...,tk distinct
cum{dN(t1),...,dN(tk)}
= qN...N (t1 ,...,tk)dt1 ...dtk

15. Stationarity.
Joint distributions,
Pr{N(I1+t)=k1 ,..., N(In+t)=kn} k1 ,...,kn integers  0
do not depend on t for n=1,2,...
Rate.
E{dN(t)=pNdt
Product density of order 2.
Pr{dN(t+u)=1 and dN(t)=1}
= [(u)pN + pNN (u)]dtdu

16. Autointensity.
Pr{dN(t+u)=1|dN(t)=1}
= (pNN (u)/pN)du u  0
= hN(u)du
Covariance density.
cov{dN(t+u),dN(t)}
= [(u)pN + qNN (u)]dtdu

20. Taking "E" again,

21. Association. Measuring? Due to chance?
Are two processes associated? Eg. t.s. and p.p.
How strongly?
Can one predict one from the other?
Some characteristics of dependence:
E(XY)  E(X) E(Y)
E(Y|X) = g(X)
X = g (), Y = h(),  r.v.
f (x,y)  f (x) f(y)
corr(X,Y)  0

22. Bivariate point process case.
Two types of points (j ,k)
Crossintensity.
Prob{dN(t)=1|dM(s)=1}
=(pMN(t,s)/pM(s))dt
Cross-covariance density.
cov{dM(s),dN(t)}
= qMN(s,t)dsdt no ()

24. Mixing.
cov{dN(t+u),dN(t)} small for large |u|
|pNN(u) - pNpN| small for large |u|
hNN(u) = pNN(u)/pN ~ pN for large |u|
 |qNN(u)|du < 
See preceding examples

25. Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
Non-negative
Unifies analyses of processes of widely varying types

26. Examples.

28. Spectral representation. stationary increments - Kolmogorov

29. Algebra/calculus of point processes.
Consider process {j, j+u}. Stationary case
dN(t) = dM(t) + dM(t+u)
Taking "E", pNdt = pMdt+ pMdt
pN = 2 pM

30. Taking "E" again,