Period of Meaning

 

Stochastic Meaning Theory 2

Period of Meaning

13^th for KARCEVSKIJ Sergej

On what there exists confirmation of meaning in word

TANAKA Akio

1 <σ additive>

Set     X

A family of subset of X     M

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

(i) X, Ø ∈M

(ii) A∈M ⇒ X╲A∈M

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

2 <Measurable space>

Set     X

Family of σ additive     M

Pair ( X, M ) is called measurable space.

3 <Measure space>

Measurable space      ( X, M )

Function over M     μ

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

(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.

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 P (A )     probability

Probability space     ( Ω, F, P ).

5 <Borel additive>

Measurable space     ( X, M ), ( Y, N )

Map     f : X→Y

Arbitrary A∈N

f ^-1 ( A ) = {x∈N ; 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     ( X, M )

Function from X to      f

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

(i) f ^-1 ( [-∞, a ] )∈M,  f ^-1 ( [-∞, a ) )∈M

(ii) f ^-1 ( [ a, ∞] )∈M,  f ^-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 < F_t+-measurable>

Countable sequence of probability space      ( Ω_n, F_n, P_n ).

9 <Random variable>

Probability space     ( Ω, F, P )

valued function over Ω     X

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

10 <Expectation (Mean)>

Probability space     ( Ω, F, P )

| X (ω) | is integrable.

Expectation of random variable EX     ∫_Ω X(ω)P(dω)

Expectation is also called mean.

11 <Covariance>

Random variable     ( X (ω) – EX )²

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

Random Variable     X, Y

Covariant    cov ( X, Y ) = E ( X- EX ) (Y-EY )    X (ω) and Y (ω) are integrable.

12 < Probability distribution >

Random variable     X

Probability     P

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

13 <Density function>

Probability distribution over R     F ( x )

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

F ( b ) - F ( a ) = ∫^b_a ρ(x)dx

14 <Gauss distribution>

m∈R^d

d×d positive definite symmetric matrix    Σ

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

Gauss distribution N ( m, Σ )      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     F_λ

Sequence of F_λ     { F_λ}_λ∈Λ

When { F_λ}_λ∈Λ satisfies the next, it is called independent on probability P.

Arbitrary finite sequence {λ₁, …, λ_n}

Arbitrary Ai∈Fλ_i  ( i = 1, 2, …, n )

P ( A₁∩A₂∩…∩A_n ) = P ( A₁ ) P ( A₂ ) …P ( A_n )   

16 <Brownian motion>

Probability space     ( Ω, F, P )

Family of R^d valued random variable     {B_t}_t≥0

When {B_t}_t≥0 satisfies the next, it is called d-dimensional Brownian motion from starting position x.

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

(ii) When 0 = t₀≤t₁≤…≤t_n, {Bt_k – Bt_k-1}ⁿ _k=1 is independent.

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

17 < F_t >

x∈R^d

d-dimensional Brownian motion that starts from x     {B_t}_t≥0

F_t is defined by the next.

F_t = σ (B_s ; s≤t )

18 <Markov time>

d-dimensional Brownian motion    ( {B_t}_t≥0, P_x )

F_t = σ (B_s ; s≤t ) , F_t+  = ∩_t >0 F_t+ε

[0, ∞] valued random variable     τ

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

(i) t≥0

(ii) {ω∈Ω ; τ (ω) ≤t }∈Ω

19 <Martingale>

Martingale is defined by the next.

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

(ii) For every t≥0, M_t is F_t+-measurable.

(iii) For every t≥0, M_t is integrable. When t≥s≥0, E ( M_t | F_s+ ) = M_s

20 <Theorem>

Continuous Martingale on F_t+     {M_t}_t≥0

T < ∞

{M_t ; o≤t ≤T } is bounded.

For bounded Markov time τ, next is brought.

EM_τ = EM₀

21 <Confirmation>

Meanings inherent in word : =  {M_t}_t≥0

All of time inherent in word : = o≤t ≤T<∞

Specific time of word that has meanings : = τ  

Specific meaning of specific time : = EMτ

Confirmation of specific meaning : = (EM_τ= EM₀)

Tokyo June 27, 2008

Sekinan Research Field of Language

www.sekinan.org