Friday 20 September 2019

Stochastic Meaning Theory Period of Meaning 12th for KARCEVSKIJ Sergej On what there exists time of meaning in word


Period of Meaning
12th for KARCEVSKIJ Sergej
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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