Monday 7 August 2023

Stochastic Meaning Theory Period of Meaning 12th for KARCEVSKIJ Sergej

 



Period of Meaning
On what there exists time of meaning in word


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) XØ M
(ii) aM  XAM
(iii) An(n=1, 2, …) n=1 AnM
XM ) is called measurable space.
Function over M     μ
When μ satisfies the next <2>(i)(ii)(iii), μ is called measure over measurable space ( XM ).
(i) μ (A)[0,]
(ii) μ (0) = 0
(iii) AnAAm = 0  (nm)
μ (n=1 An) = Σn=1 μ (A)
XM, μ ) is called measure space.
When measure space satisfies the next <3>(i), it is called complete measure space.
(i) AMμ (A) = 0  BA, 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     that is called event
Function over F   P 
Measure for AF     () that is called probability      

4
Probability space     ( ΩFP )
valued function over Ω     X
When X is F- measurable, it is called random variable.
When value of measurable space (SM) is not  but S, variable is called S valued random variable.
Expectation of random variable over ( ΩFP ) : = Ω X(ω)P()     EX
Family of subsets of Ω     A}n=1
When { A}n=1 satisfies the next <1>(i)(ii),  it is called countable decomposition of Ω.
(i)  A A­= Ø  ( ≠ m )
(ii) n=1 AΩ

5
Almost countable set     that has σ-field
Separable space     ( ΩF )
Sequence of S valued random variable      {Xn}n=0
Sub-σ-field of F     Fn  : = σ ( Xk ; 0 1)
xS
(x, y) 1
When Σ p(xy) = 1, x is satisfied, p is called transition probability.
Family of probability measure     {Pz}zS
When {Xn}n=1 and {Pz}zis satisfies the next <2>(i)(ii) for bounded function over S, they are called Markov chain that has transition probability p.
<2>
(i)  PX = x ) = 1
(ii) Ex ( (Xn+1 ) | F) =Σ (Xny) f(y) a. s. Px
<2>(ii) is called Markov property.

6
n = 1, 2, …
Measurable map : Ω  Ω
Shift of pass     θ
θnθm = θn+m
Xn(θ) : = Xn+m( ω )
Markov chain that has shift θn     {Xn}n=0
F : = σ Xn = 1, 2, … )
Bounded function over F-measurable Ω     f
Ex ( (θnω ) | F) = Exn xn( f )   a, s Px

7
Space that has Markov chain     {Xn}n=0
Transition probability    p ( xy ) : =Σx1, x2, …, xp xx1 ) p ( x1x2)…p ( xny )
N0 : = N{0}
N (x) = { nNp ( xy ) > 0 }
Greatest common divisor of N (x)     dx
dis called period of xS.
When xis dx =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/ 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]

Tokyo June 22, 2008

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