Place of Meaning
For Aurora Theory especially for Dictron and Aurora <Language is aurora dancing above us.>
1
Sample space Ω
Element of Ω ω
ω is called sample point.
Subset C⊂Ω
C is called event.
C = Ω is all event.
C = ø is null event.
1-1
Valued space X
Index space I
Space Ω = X I
Element ω = {ai ; i∈I, ai∈X}
1-2
Ω is finite. |Ω| =m <∞
All the subsets of Ω F
F is all of event C.
F consists of 2m number events.
Family of subsets of Ω G
G that satisfies the next is called additive family.
(i) Ω∈G
(ii) C∈G ⇒ CC∈G
(iii) C1, C2, …, Ck∈G ⇒ ⋃k i =1∈G
Complement of C CC
1-3
Family of subsets of Ω F
G that satisfies the next is called perfect additive family.
(i) F is additive family.
(ii) C1, C2, …, Ck∈F ⇒ ⋃∞ i =1∈F
1-4
Perfect additive family F
Measurable space (Ω, F)
1-5
Ω is finite.
Arbitrary real function f = f (ω)
f is called random variable.
1-6
Arbitrary sub-perfect additive family F0 ∈F
Arbitrary a, b a ≤b
When a, b satisfy the next, it is called what random variable ε = f (ω) is F0- measurable.
{ω | a ≤f (ω)≤b}∈F0
1-7
Function defined over F P
P that satisfies the next is called probability.
(i) For arbitrary C∈F, P ( C ) ≥ 0
(ii) P (Ω) = 1
(iii) i = 1, 2, … When Ci∈F and ci∩cj = ø, P ( ⋃ ∞ i=1Ci ) = ∑∞ i=1P (Ci ).
P (C) is called probability of event C.
1-8
(Ω, F, P) is called probability space.
2
2-1
Probability space (Ω, F, P)
Event A∈F, B∈F
P (B)>0
A’s conditional probability on event B is defined by the next.
P ( A | B ) = 
When event A and B satisfy the next, they are called independent.
P(A ∩B) = P(A)・P(B)
2-2
Sub-perfect additive family F1, F2
Arbitrary C1∈F1, C2∈F2
When C1 and C2 satisfy the next, F1 and F2 are called independent.
P(C1∩C2) = P(C1)・P(C2)
Perfect additive family F
Finite family of F’s sub-perfect additive family. F1, F2, …, Fn
When C1 ,C2, …, Cn satisfy the next, Fi (1≤i ≤n) is called independent.
P(C1∩C2∩…∩Cn) = P(C1)・P(C2)…P(Cn)
2-3
Family of n-number random variable η1 =f1(ω), …, ηn = fn(ω)
Element of Borel sets’ family C1, …, Cn
When η1, …, ηn satisfies the next, η1, …, ηn is called independent random variable on C1, …, Cn.
P{ η1 =f1(ω)∈C1, …, ηn = fn(ω)∈Cn } = ∏ni =1 P{ fi(ω)∈Ci }
When η1, …, ηn has density function p1(x), …, pn(x), η1, …, ηn satisfies the next.
P{ a1≤η1≤b1, …, an≤ηn≤bn } = ∏ni =1∫bkak pk(x)dx
<Theorem>
Independent random variable η1,η2, …, ηn
1≤i ≤n
Eηi < ∞
There exists E(η1・η2・・・ ηn ) and η1,η2, …, ηn = Eη1 …,Eηn is formed.
3
3-1
Matrix P = [pij] (i, j = 1, 2,…, n)
P that satisfies the next is called stochastic matrix.
(i) pij≥0
(ii) ∑nj =1 pij = 1 (i, j = 1, 2,…, n)
3-2
Probability space (Ω, F, P)
Sample point ω
Ω = {ωi}
Cω := {ω}
Probability of ω p (ω) = P(Cω) = P ({ω})
The set of numbers that satisfies the next is called probability distribution.
(i) p (ω)≥0
(ii) ∑ωp (ω) = 1
3-3
Space of sample point ω = (ω0, ω1, …, ωn) Ω
State space X
0 ≤ i ≤ n
ωi ∈X = {x(1), x(2), …, x(r)}
Initial distribution 
Probability matrix P(1), P(2), …, P(n)
Probability distribution over Ω P
X , and P(1), P(2), …, P(n) that satisfies the next is called Markov chain.
and P(1), P(2), …, P(n) that satisfies the next is called Markov chain.p (ω) = μω0 . μω0ω1(1) …μωn-1ωn(n)
Markov chain that does not depend on k(1≤k≤n) is called invariant Markov chain..
3-4
Invariant Markov chain P
Conditional probability P(ωs+l = (x(j) | ωl =x(i))
P(ωl = x(i))>0
P(ωs+l = (x(j) | ωl =x(i)) = p (s)ij
p (s)ij is called s class transitive probability.
3-5
Matrix P
P has a certain s0.
For arbitrary i, j p(s0)ij>0, P is called ergodic.
3-6
<Ergodic theorem>
Ergodic transitive matrix P
When Markov chain that has P is given, there exists only one probability distribution π = (π1, …, πr)that satisfies the next.
(i) πP = π
(ii) lims→∞p(s)ij = πj
4
4-1
Point x = (x1, …, xd) -∞<xi <∞
Integer 1≤i≤d
Lattice Zd
Random walk over Zd Markov chain at state space X = Zd
Random distribution over Zd p = {pz | z∈Zd}
p that satisfies the next is called to be uniform in space.
Pxy = Py-x
4-2
Locus of random walk ω = (ω0, ω1, …, ωk)
Random walk that starts from the origin ω0 = 0, pωi -ωi-1 >0
All ωs that first return to the origin toward which ω happens to be at k th Ω(k)
k>0
ω∈Ω(k)
p (ω) = pω１-ω0・・・ pωk-ωk-1
f k = ∑ω∈Ω(k) p (ω)
f 0 := 0
Random walk that satisfies the next is called to be recurrent.
∑ω∈Ω(k) f k = 1
Random walk that satisfies the next is called to be transient.
∑ω∈Ω(k) f k < 1
4-3
Arbitrary bounded sequence {an}
Generating function of {an} ∑k≥0 anzn
4-4
Generating function F(z) = ∑k≥0 f k zk P(z) = ∑k≥0 pk zk
pk = ∑ki = 0fi .pk-i
p0 = 1
F(z) = 1 – 1/ P(z)
From Abel’s theorem,
∑∞k = 1 f k = 1- lim z→1(1/ P(z) )
When ∑∞k = 0 pk = ∞ , lim z→1(1/ P(z) ) = 1/ ∑∞k = 0pk = 0
Random walk that is only ∑∞k = 0 pk = ∞ is recurrent.
4-5
e : = ∑z∈Zd zpz
Random walk that satisfies the next is called simple random walk.
(i)Unit coordinate vector e1, e2, …, ed
(ii-1)When y = ±es (1≤s≤d) , py-x = 1/2d.
(ii-2)When y ≠±es (1≤s≤d) , py-x = 0.
<Polya’s theorem>
When d = 1, 2 , simple random walk is recurrent.
When d ≥3, simple random walk is transient.
4-6
Unit vector νn = ωn / ||ωn||
Unit vector is distributed on unit sphere by being uniform in space.
4-7
From 4-1
Word : = x = (x1, …, xd) -∞<xi <∞
From 4-5
Language space : = d ≥3 and transient
From 4-6
Sentence : = νn
[References]
<On vector, sphere and Language>
<More details on Aurora Theory group>
Tokyo July 11, 2008
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