Tuesday, 28 February 2023

Place of Meaning




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     CΩ
C is called event.
C = Ω is all event.
C = ø is null event.
1-1
Valued space     X
Index space     I
Space     Ω I
Element     ω = {ai ; iIaiX}
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
that satisfies the next is called additive family.
(i)  ΩG
(ii)  C⇒ CCG
(iii)  C1C2, …, Ck i =1G
Complement of C     CC
1-3
Family of subsets of Ω   F
that satisfies the next is called perfect additive family.
(i) F is additive family.
(ii) C1C2, …, CkF   i =1F  
1-4
Perfect additive family     F
Measurable space     (Ω, F)
1-5
Ω is finite.
Arbitrary real function     f (ω)
f is called random variable.
1-6
Arbitrary sub-perfect additive family     FF
Arbitrary ab     a b
When ab satisfy the next, it is called what random variable ε f (ω) is F0- measurable.
{ω | f (ω)b}F0
1-7
Function defined over F     P
P that satisfies the next is called probability.
(i) For arbitrary CFP ( C )  0
(ii) P (Ω) = 1
(iii) i = 1, 2, …   When Ciand cicø, P (   i=1Ci ) = ∑ i=1(Ci ).
(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     AF, BF
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(B) = P(A)P(B)
2-2
Sub-perfect additive family       F1F2
Arbitrary C1F1, C2F2
When Cand C2 satisfy the next, Fand Fare called independent.
P(C1C2) = P(C1)P(C2)
Perfect additive family     F
Finite family of F’s sub-perfect additive family. F1F2, …, Fn
When C1 ,C2, …, Cn satisfy the next, F(1i n) is called independent.
P(C1C2Cn) = 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, …, ηsatisfies the next, η1, …, ηis called independent random variable on C1, …, Cn.
P{ η1 =f1(ω)C1, …, ηn = fn(ω)Cn } = n=1 P{ fi(ω)C}
When η1, …, ηhas density function p1(x), …, pn(x), η1, …, ηsatisfies the next.
P{ a1η1b1anηnbn } = n=1bkak pk(x)dx
<Theorem>
Independent random variable     η12…, ηn   
1n
Eηi < 
There exists E(η1η2・・・ ηn ) and  η12…, 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) pij0
(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) (ω)0
(ii) ω(ω) = 1
3-3
Space of sample point ω = (ω0, ω1, …, ωn)      Ω
State space     X
≤ i ≤ n
ωi = {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.
(ω) = μω0 . μω0ω1(1) …μωn-1ωn(n)
Markov chain that does not depend on k(1kn) is called invariant Markov chain..
3-4
Invariant Markov chain     P
Conditional probability     P(ωs+l = (x(jω=x(i))
P(ωx(i))>0
P(ωs+l = (x(jω=x(i)) = p (s)ij
p (s)ij is called class transitive probability.
3-5
Matrix    P
P has a certain s0.
For arbitrary ij 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) limsp(s)ij = πj

4
4-1
Point     x = (x1, …, xd)  -<xi <
Integer     1id   
Lattice     Zd
Random walk over Zd     Markov chain at state space Zd
Random distribution over Zd     = {pz | zZd}
p that satisfies the next is called to be uniform in space.
Pxy = Py-x
4-2
Locus of random walk     ω = (ω0ω1, …, ωk)
Random walk that starts from the origin     ω0 = 0, -ωi-1 >0
All ωs that first return to the origin toward which ω happens to be at th      Ω(k)
k>0
ωΩ(k)
(ω) = -ω0・・・ k-ωk-1
k ωΩ(k) p (ω)
f 0 := 0
Random walk that satisfies the next is called to be recurrent.
ωΩ(kk = 1
Random walk that satisfies the next is called to be transient.
ωΩ(kk < 1
4-3
Arbitrary bounded sequence     {an}
Generating function of {an}     kanzn
4-4
Generating function    F(z) = kk zk     P(z) = k0 pk zk
pk = ki = 0fi .pk-i
p0 = 1
F(z) = 1 – 1/ P(z)
From Abel’s theorem,
k = 1 k = 1- lim z1(1/ P(z) )
When k = 0 pk  , lim z1(1/ P(z) ) = 1/ k = 0p= 0
Random walk that is only k = 0 pk ∞ is recurrent.
4-5
e : = zZzpz
Random walk that satisfies the next is called simple random walk.
(i)Unit coordinate vector     e1e2, …, ed
(ii-1)When y = ±es (1sd) , py-x = 1/2d.
(ii-2)When y ≠±es (1sd) , py-x = 0.
<Polya’s theorem>
When d = 1, 2 , simple random walk is recurrent.
When 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 : = 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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