Friday, 16 May 2025

Stochastic Meaning Theory 2 Period of Meaning 13th for KARCEVSKIJ Sergej On what there exists confirmation of meaning in word

 Stochastic Meaning Theory 2

 

Period of Meaning

13th for KARCEVSKIJ Sergej

On what there exists confirmation of meaning in word

 

TANAKA Akio

 

1 <σ additive>

Set     X

A family of subset of     M

When M satisfies the next, it is called σ additive.

(i) XØ M

(ii) AM  XAM

(iii) An(n=1, 2, …) n=1 AnM

 

2 <Measurable space>

Set     X

Family of σ additive     M

Pair ( XM ) is called measurable space.

 

3 <Measure space>

Measurable space      ( XM )

Function over M     μ

When μ is satisfies the next, it is called measure.

(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.

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

 

4 <Probability space>

Measure space in which all the measures are 1 is called probability space.

Set     Ω

Element of Ω     ω

σ-field      F

Element of F     A

Function over F   P 

Measure ()     probability

Probability space     ( ΩFP ).

 

5 <Borel additive>

Measurable space     ( XM ), ( YN )

Map     XY

Arbitrary AN

f -1 ( A ) = {xN ; f (x)A }M

Map f is called M-N measurable.

A family of subsets of X     U

σ ( U ) = ( M ; M is σ additive that contains U )

σ ( U ) is also notated B ( X ) that is called Borel σ additive.

= [-, +]

Borel σ additive of  is notated B(.

Element of Borel σ additive is called Borel set.

 

6 < M-B(measurable>

Measurable space     ( XM )

Function from X to      f

When f satisfies one of the next, it is called M-B(measurable.

(i) -1 ( [-] )M,  -1 ( [-a ) )M

(ii) -1 ( [ a] )M,  -1 ( (a] )M

 

7 <F-measurable>

When function f : X is M-B(measurable, it is called M-measurable function, that is generally notated F-measurable.

 

8 < Ft+-measurable>

Countable sequence of probability space      ( ΩnFnPn ).

 

9 <Random variable>

Probability space     ( ΩFP )

valued function over Ω     X

When X is Ft+-measurable, X is called random variable.

 

10 <Expectation (Mean)>

Probability space     ( ΩFP )

| X (ω) | is integrable.

Expectation of random variable EX     Ω X(ω)P()

Expectation is also called mean.

 

11 <Covariance>

Random variable     ( X (ω) – EX )2

Variance     Expectation of ( X (ω) – EX )2    

Random Variable     XY

Covariant    cov ( XY ) = E ( XEX ) (Y-EY )    X (ω) and Y (ω) are integrable.

 

12 < Probability distribution >

Random variable     X

Probability     P

Probability distribution function     F (x) = P ( X)

 

13 <Density function>

Probability distribution over R     F x )

Function ρ(x) satisfies the next, it is called density function for F.

b ) - F a ) = bρ(x)dx

 

14 <Gauss distribution>

mRd

d×d positive definite symmetric matrix    Σ

Density function over Rfor Σ     ( 2π )-d/2 (det Σ )-1/2exp{-1/2 (Σ-1 (x-m), x-m ) }

Gauss distribution N ( mΣ )      distribution that has ( 2π )-d/2 (det Σ )-1/2exp{-1/2 (Σ-1 (x-m), x-m ) }

 

15 <Independent>

Set    Λ

Element of Λ     λ

Sub-family of σ additive F     Fλ

Sequence of Fλ     { Fλ}λΛ

When { Fλ}λΛ satisfies the next, it is called independent on probability P.

Arbitrary finite sequence {λ1, …, λn}

Arbitrary AiFλi  = 1, 2, …, )

P ( A1A2∩…∩An ) = AA2 ) P ( A  

 

16 <Brownian motion>

Probability space     ( ΩFP )

Family of Rd valued random variable     {Bt}t0

When {Bt}tsatisfies the next, it is called d-dimensional Brownian motion from starting position x.

(i) B0 = x at probability 1 and Bis continuous on t.

(ii) When 0 = t0t1tn, {Btk – Btk-1}n k=1 is independent.

(iii) When 0s<tbt – Bs is mean 0, Gauss distribution of covariant matrix ( t-s )I.

 

17 < F>

xRd

d-dimensional Brownian motion that starts from x     {Bt}t0

Ft is defined by the next.

Fσ (Bst )

 

18 <Markov time>

d-dimensional Brownian motion    ( {Bt}t0Px )

Fσ (Bst ) , Ft+  = ∩>0 Ft+ε

[0, ] valued random variable     τ

When τ satisfies the next, it is called Markov time on Ft+ε.

(i) t0

(ii) {ωΩ τ (ω) }Ω

 

19 <Martingale>

Martingale is defined by the next.

(i)  {Mt} is continuous at probability 1.

(ii) For every t0, Mt is Ft+-measurable.

(iii) For every t0, Mt is integrable. When ts0, E ( Mt Fs) = Ms

 

20 <Theorem>

Continuous Martingale on Ft+     {Mt}t0

{M; o} is bounded.

For bounded Markov time τ, next is brought.

EMτ = EM0

 

21 <Confirmation>

Meanings inherent in word : =  {Mt}t0

All of time inherent in word : = oT<∞

Specific time of word that has meanings : = τ  

Specific meaning of specific time : = EMτ

Confirmation of specific meaning : = (EMτEM0)

 

Tokyo June 27, 2008

Sekinan Research Field of Language

www.sekinan.org