Stochastic Meaning Theory Period of Meaning 12th for KARCEVSKIJ Sergej On what there exists time of meaning in word / June 22, 2008

Stochastic Meaning Theory

Period of Meaning

12^th for KARCEVSKIJ Sergej

On what there exists time of meaning in word

TANAKA Akio

1

Set     X

Family of subsets of X     M

When M satisfies the next <1>(i)(ii)(iii), M is called σ-field.

<1>

(i) X, Ø ∈M

(ii) a∈M ⇒ X╲A∈M

(iii) A_n∈M (n=1, 2, …) ⇒∪^∞_n=1 A_n∈M

( X, M ) is called measurable space.

Function over M     μ

When μ satisfies the next <2>(i)(ii)(iii), μ is called measure over measurable space ( X, M ).

(i) μ (A)∈[0,∞]

(ii) μ (0) = 0

(iii) A_n∈M , A_n ∩A_m = 0  (n≠m)

μ (∪^∞_n=1 A_n) = Σ^∞_n=1 μ (A)

( X, M, μ ) is called measure space.

When measure space satisfies the next <3>(i), it is called complete measure space.

(i) A∈M, μ (A) = 0 ⇒ B⊂A, μ (B) = 0

<2>(iii) is called complete additive or σ additive.

2

Measure space that is all the measure is 1 is called probability space.

Measure that all the measure is 1 is called probability measure.

3

Set     Ω that is called whole possibility

Element of Ω     ω that is called sample point

σ-field      F

Element of F     A that is called event

Function over F   P 

Measure for A∈F     P (A ) that is called probability      

4

Probability space     ( Ω, F, P )

valued function over Ω     X

When X is F- measurable, it is called random variable.

When value of measurable space (S, M) is not  but S, variable is called S valued random variable.

Expectation of random variable over ( Ω, F, P ) : = ∫_Ω X(ω)P(dω)     EX

Family of subsets of Ω     { A_n }^∞_n=1

When { A_n }^∞_n=1 satisfies the next <1>(i)(ii),  it is called countable decomposition of Ω.

(i)  A_n ∩ A_m ­= Ø  ( n ≠ m )

(ii) ∪^∞_n=1 A_n = Ω

5

Almost countable set     S that has σ-field

Separable space     ( Ω, F )

Sequence of S valued random variable      {X_n}^∞_n=0

Sub-σ-field of F     F_n  : = σ ( X_k ; 0 ≤k ≤1)

x, y ∈S

0 ≤p (x, y) ≤1

When Σ _y ∈S p(x, y) = 1, x∈S  is satisfied, p is called transition probability.

Family of probability measure     {P_z}_z∈S

When {X_n}^∞_n=1 and {P_z}_z∈S is satisfies the next <2>(i)(ii) for bounded function over S, they are called Markov chain that has transition probability p.

<2>

(i)  P_z ( X₀  = x ) = 1

(ii) E_x ( f (X_n+1 ) | F_n ) =Σ _y ∈S p (X_n, y) f(y) a. s. P_x

<2>(ii) is called Markov property.

6

n = 1, 2, …

Measurable map : Ω → Ω

Shift of pass     θ

θ_n・θ_m : = θ_n+m

X_n(θ_mω) : = X_n+m( ω )

Markov chain that has shift θ_n     {X_n}^∞_n=0

F_∞ : = σ ( X_n ; n = 1, 2, … )

Bounded function over F_∞-measurable Ω     f

E_x ( f (θ_nω ) | F_n ) = E_xn xn( f )   a, s P_x

7

Space that has Markov chain     {X_n}^∞_n=0

Transition probability    p ( x, y ) : =Σ_{x1, x2, …, xn} p ( x, x₁ ) p ( x₁, x₂)…p ( x_n, y )

N₀ : = N∪{0}

N (x) = { n∈N₀ ; p ( x, y ) > 0 }

Greatest common divisor of N (x)     d_x

d_x is called period of x∈S.

When x∈S is d_x =1, it is called aperiodic.

8

In early work, time within inner structure of word was considered.

The paper is “On Time Property Inherent in Characters” in which Chinese character /geng/ 亙that means eternity in English is taken.

This character is supposed to have period of Markov chain.

Meaning elements is supposed to be sequence of random variable.

[Reference]

OnTime Property Inherent in Characters / Hakuba March 28, 2003

Early work

Tokyo June 22, 2008

Sekinan Research Field of Language