Stochastic Meaning Theory 3 Place of Meaning For Aurora Theory especially for Dictron and Aurora, Language is aurora dancing above us. / July 11, 2008

Stochastic Meaning Theory 3

Place of Meaning

For Aurora Theory especially for Dictron and Aurora <Language is aurora dancing above us.>

TANAKA Akio

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 = {a_i ; i∈I, a_i∈X}

1-2

Ω is finite.     |Ω| =m <∞

All the subsets of Ω     F

F is all of event C.

F consists of 2^m number events.  

Family of subsets of Ω    G

G that satisfies the next is called additive family.

(i)  Ω∈G

(ii)  C∈G ⇒ C^C∈G

(iii)  C₁, C₂, …, C_k∈G ⇒ ⋃^k _i =1∈G

Complement of C     C^C

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, …, C_k∈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 C_i∈F and c_i∩c_j = ø, P ( ⋃ ^∞ _i=1C_i ) = ∑^∞ _i=1P (C_i ).

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∈F_1, 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, …, F_n

When C_{1 ,}C₂, …, C_n satisfy the next, F_i (1≤i ≤n) is called independent.

P(C₁∩C₂∩…∩C_n) = P(C₁)・P(C₂)…P(C_n)

2-3

Family of n-number random variable     η₁ =f₁(ω), …, η_n = f_n(ω)

Element of Borel sets’ family     C₁, …, C_n

When η₁, …, η_n satisfies the next, η₁, …, η_n is called independent random variable on C₁, …, C_n.

P{ η₁ =f₁(ω)∈C₁, …, η_n = f_n(ω)∈C_n } = ∏ⁿ_i =1 P{ f_i(ω)∈C_i }

When η₁, …, η_n has density function p₁(x), …, p_n(x), η₁, …, η_n satisfies the next.

P{ a₁≤η₁≤b₁, …, a_n≤η_n≤b_n } = ∏ⁿ_i =1∫^bk_ak p_k(x)dx

<Theorem>

Independent random variable     η₁,η₂, …, η_n   

1≤i ≤n

Eη_i < ∞

There exists E(η_1・η_2・・・ η_n ) and  η₁,η₂, …, η_n  = Eη₁ …,Eη_n is formed.  

3

3-1

Matrix     P = [p_ij]  (i, j = 1, 2,…, n)

P that satisfies the next is called stochastic matrix.

(i) p_ij≥0

(ii) ∑ⁿ_j =1 p_ij = 1  (i, j = 1, 2,…, n)

3-2

Probability space     (Ω, F, P)

Sample point     ω

Ω = {ω_i}

C_ω := {ω}

Probability of ω    p (ω) = P(C_ω) = 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, …, ω_n)      Ω

State space     X

0 ≤ i ≤ n

ω_i ∈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.

p (ω) = μω₀ . μω₀ω₁(1) …μω_n-1ω_n(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(ω_s+l = (x^(j) | ω_l =x⁽ⁱ⁾)

P(ω_l = x⁽ⁱ⁾)>0

P(ω_s+l = (x^(j) | ω_l =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, …, π_r)that satisfies the next.

(i) πP = π

(ii) lim_s→∞p^(s)_ij = π_j

4

4-1

Point     x = (x1, …, x_d)  -∞<x_i <∞

Integer     1≤i≤d   

Lattice     Z^d

Random walk over Z^d     Markov chain at state space X = Z^d

Random distribution over Z^d     p = {p_z | z∈Z^d}

p that satisfies the next is called to be uniform in space.

Pxy = Py-x

4-2

Locus of random walk     ω = (ω₀, ω₁, …, ω_k)

Random walk that starts from the origin     ω₀ = 0, pω_i -ω_i-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) = pω_１-ω_0・・・ pω_k-ω_k-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     {a_n}

Generating function of {a_n}     ∑_k≥0 a_nzⁿ

4-4

Generating function    F(z) = ∑_k≥0 f _k z^k     P(z) = ∑_k≥0 p_k z^k

p_k = ∑^k_{i = 0}f_i .p_k-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} p_k = ∞ , lim _z→1(1/ P(z) ) = 1/ ∑^∞_{k = 0}p_k = 0

Random walk that is only ∑^∞_{k = 0} p_k = ∞ is recurrent.

4-5

e : = ∑_z∈Z^d zp_z

Random walk that satisfies the next is called simple random walk.

(i)Unit coordinate vector     e₁, e₂, …, e_d

(ii-1)When y = ±e_s (1≤s≤d) , p_y-x = 1/2d.

(ii-2)When y ≠±e_s (1≤s≤d) , p_y-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     ν_n = ω_n / ||ω_n||

Unit vector is distributed on unit sphere by being uniform in space.

4-7

From 4-1

Word : = x = (x1, …, x_d)  -∞<x_i <∞

From 4-5

Language space : = d ≥3 and transient

From 4-6

Sentence : = ν_n

[References]

<On vector, sphere and Language>

Aurora Theory / Dictron as Language Quantum / Tokyo October 1, 2006

Aurora Theory / Aurora and Riemann Sphere / Tokyo October 2, 2006

Aurora Theory / Dictron, Time and Symmetry / Tokyo October 6, 2006

Aurora Theory / Aurora Plane / Tokyo October 14, 2006

<More details on Aurora Theory group>

Aurora Theory

Aurora Time Theory

Language and Spacetime

Tokyo July 11, 2008

Sekinan Research Field of Language