Sekinan Table
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SRFL News: At least three elements for language universals 2015
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Tuesday, 6 December 2022
Stochastic Meaning Theory 5 Language as the Brownian motion
Language as the Brownian motion
[A]
1
Abstractive space Ω
σ additive family that consists of subset of Ω F
Measure that is defined over F P
P satisfies P (Ω ) = 1.
P probability measure over ( Ω, F )
Ω sample space
( Ω, F , P ) Probability space
Element of Ω sample ω
Element of F event A
Probability that event A occurs probability P ( A )
Real number valued Borel measurable function over Ω random variable X = X ( ω )
Random variable is integrable.
Mean (Expectation) of X E[X] = ∫Ω X ( ω ) P ( dω )
2
Measurable space ( S, S )
X : ( Ω, F ) → ( S, S )
X is measurable.
X S-value random variable.
Random variable X1, …, Xd
X : = (X1, …, Xd ) Rd-value random variable
3
Rd-value random variable X
E[X i 2] < ∞
E[(X - E[X])2] variance
4
S-value random variable. X
PX : = P ( X ∈A ), A∈S distribution
5
Real number space R
Borel set family over R B ( R )
Probability measure over ( R, B ( R ) ) μ
6
Rd-value random variable X
ψX (ξ ) : = E[eiξ・X], ξ∈Rd characteristic function
7
Lebesgue measure dx
Mean m∈R
Variance v >0
Measure over R μ ( dx ) = e -(x-m)2 / 2v dx /
Gauss distribution ( normal distribution)
Gauss distribution ( normal distribution)8
(2p – 1 ) !! : = (2p – 1 ) ・(2p – 3 ) … 3・1
E[X2p] = (2p – 1 ) !! v p moment of X
9
Event A, B∈F
When (A∩B) = P(A) P(B), A and B are independent each other.
10
Integrable and independent random variable X, Y
Product XY integrable
E[XY] = E[X]E[Y]
11
Time t
t ∈[0, ∞)
Family of Rd-value random variable ≥ X = ( Xt ) t ≥ 0 d-dimensional stochastic process
∀ω∈Ω
When Xt (ω) is continuous as function of t., d-dimensional stochastic process is called to be continuous.
12
σ additive family Ft
Ft ⊂F
0 ≤ s ≤ t
F s ⊂Ft
(Ft ) = (Ft ) t ≥ 0 increase information system
13
d-dimensional stochastic process X = ( Xt ) t ≥ 0
∀t ≥ 0
Xt : Ω → Rd is Ft – measurable.
X = ( Xt ) t ≥ 0 is (Ft ) – adapted.
14
Mapping ( t, ω) ∈([0, ∞)×Ω, B([0, ∞)]×F) ↦ Xt ( ω) ∈( Rd, B ( Rd ) )
When the mapping is measurable, X = ( Xt ) t ≥ 0 is called to be measurable.
15
X = ( Xt )
Ft0 = Ft0,X : = σ ( XS ; s≤t )
16
Probability space ( Ω, F , P )
Stochastic process defined over ( Ω, F , P ) (Bt)t ≥ 0 = (Bt(ω)) t ≥ 0
(Bt)t ≥ 0 that satisfies the next, it is called Brownian motion.
(i) P ( B0 = 0 ) = 1
(ii) For ∀ω∈Ω, Bt (ω) is continuous on t.
(iii) For 0 = t0<∀t1<…<tn, ∀n∈N, {Bti-Bti-1} satisfies the next.
a) {Bti-Bti-1} are independent each other.
b) {Bti-Bti-1} are followed by mean 0 and variance ti-ti-1 of Gauss distribution.
17
(Existence theorem)
Over adequate probability space, there exists Brownian motion.
18
Ω = W0
F = B ( W0 )
Brownian motion has the next.
(i)Bt ( w ) = Wt
(ii) w = ( wt ) t ≥0 ∈W 0
Measure over ( W0, B ( W0 ) ) P
P is called Wiener measure.
19
d-dimensional Brownian motion B = ( Bt ) t ≥ 0
d×d orthogonal matrix A
ABt d-dimensional Brownian motion
Sphere S : = δ B (0, r), B (0, r) = {|x| ≤ r }
Hitting time σS (ω) : = inf{t >0; Bt ∈S }
Hitting place BσS (ω)
Distribution of BσS (ω) uniform stochastic measure
20
d-dimensional Brownian motion B = ( Bt ) t ≥ 0
x∈Rd
Brownian motion from x ( x + Bt ) t ≥ 0
W d = B ( W d )
Space (W d, W d )
Distribution over (W d, W d ) Px
Mean on Px Ex [ ・ ]
Probability space (W d, W d , Px )
Stochastic process over (W d, W d , Px ) Bt ( w ), w∈W d ; Bt ( w ) = wt
Sub σ additive family of W d Ft0 =σ (Bs ; s≤t ) , Ft = Ft0 ⋁ N, t≥0 ; N : = {N∈W d ; Px (N) = 0, ∀x ∈Rd }
Ft* = Ft+ : = ∩s>t Fs
Shift operator over W d θs : W d → W d , s≥0 ; (θs (w) ) t : = wt+s
Bt ∘ θs = Bt+s
21
(Markov property)
∀x∈Rd
∀s≥0
∀Y = Y (w) : W d –measurable bounded function over W d
Ex[Y∘θs ・1A] = Ex[EBs(w)[Y]∘θs ・1A] , ∀A∈F s
By conditional mean
Ex[Y∘θs | Fs] (w) = EBs(w)[Y
Px-a.s.w
22
(Blumenthal’s 0-1 law)
When A∈F0 ( = F0* ), Px (A) = 0 or 1
23
Random variant of 1-dimensional Brownian motion starting from the origin B
σ (0,∞) : = inf {t >0; Bt∈(0,∞) }
A = {σ(0,∞) = 0 }
A ∈F0*
P (σ(0,∞) = 0 ) = 0 or 1
t↓0
P (σ(0,∞) = 0 ) = 1
From symmetry of Brownian motion Bt = -Bt
[B]
Language that has Brownian motion LB
LB has actual language and imaginary language.
[References]
To be continued
Tokyo August 12, 2008
Monday, 5 December 2022
Stochastic Meaning Theory 3 Place of Meaning
Place of Meaning
For Aurora Theory especially for Dictron and Aurora <Language is aurora dancing above us.>
1
Sample space Ω
Element of Ω ω
ω is called sample point.
Subset C⊂Ω
C is called event.
C = Ω is all event.
C = ø is null event.
1-1
Valued space X
Index space I
Space Ω = X I
Element ω = {ai ; i∈I, ai∈X}
1-2
Ω is finite. |Ω| =m <∞
All the subsets of Ω F
F is all of event C.
F consists of 2m number events.
Family of subsets of Ω G
G that satisfies the next is called additive family.
(i) Ω∈G
(ii) C∈G ⇒ CC∈G
(iii) C1, C2, …, Ck∈G ⇒ ⋃k i =1∈G
Complement of C CC
1-3
Family of subsets of Ω F
G that satisfies the next is called perfect additive family.
(i) F is additive family.
(ii) C1, C2, …, Ck∈F ⇒ ⋃∞ i =1∈F
1-4
Perfect additive family F
Measurable space (Ω, F)
1-5
Ω is finite.
Arbitrary real function f = f (ω)
f is called random variable.
1-6
Arbitrary sub-perfect additive family F0 ∈F
Arbitrary a, b a ≤b
When a, b satisfy the next, it is called what random variable ε = f (ω) is F0- measurable.
{ω | a ≤f (ω)≤b}∈F0
1-7
Function defined over F P
P that satisfies the next is called probability.
(i) For arbitrary C∈F, P ( C ) ≥ 0
(ii) P (Ω) = 1
(iii) i = 1, 2, … When Ci∈F and ci∩cj = ø, P ( ⋃ ∞ i=1Ci ) = ∑∞ i=1P (Ci ).
P (C) is called probability of event C.
1-8
(Ω, F, P) is called probability space.
2
2-1
Probability space (Ω, F, P)
Event A∈F, B∈F
P (B)>0
A’s conditional probability on event B is defined by the next.
P ( A | B ) = 
When event A and B satisfy the next, they are called independent.
P(A ∩B) = P(A)・P(B)
2-2
Sub-perfect additive family F1, F2
Arbitrary C1∈F1, C2∈F2
When C1 and C2 satisfy the next, F1 and F2 are called independent.
P(C1∩C2) = P(C1)・P(C2)
Perfect additive family F
Finite family of F’s sub-perfect additive family. F1, F2, …, Fn
When C1 ,C2, …, Cn satisfy the next, Fi (1≤i ≤n) is called independent.
P(C1∩C2∩…∩Cn) = P(C1)・P(C2)…P(Cn)
2-3
Family of n-number random variable η1 =f1(ω), …, ηn = fn(ω)
Element of Borel sets’ family C1, …, Cn
When η1, …, ηn satisfies the next, η1, …, ηn is called independent random variable on C1, …, Cn.
P{ η1 =f1(ω)∈C1, …, ηn = fn(ω)∈Cn } = ∏ni =1 P{ fi(ω)∈Ci }
When η1, …, ηn has density function p1(x), …, pn(x), η1, …, ηn satisfies the next.
P{ a1≤η1≤b1, …, an≤ηn≤bn } = ∏ni =1∫bkak pk(x)dx
<Theorem>
Independent random variable η1,η2, …, ηn
1≤i ≤n
Eηi < ∞
There exists E(η1・η2・・・ ηn ) and η1,η2, …, ηn = Eη1 …,Eηn is formed.
3
3-1
Matrix P = [pij] (i, j = 1, 2,…, n)
P that satisfies the next is called stochastic matrix.
(i) pij≥0
(ii) ∑nj =1 pij = 1 (i, j = 1, 2,…, n)
3-2
Probability space (Ω, F, P)
Sample point ω
Ω = {ωi}
Cω := {ω}
Probability of ω p (ω) = P(Cω) = P ({ω})
The set of numbers that satisfies the next is called probability distribution.
(i) p (ω)≥0
(ii) ∑ωp (ω) = 1
3-3
Space of sample point ω = (ω0, ω1, …, ωn) Ω
State space X
0 ≤ i ≤ n
ωi ∈X = {x(1), x(2), …, x(r)}
Initial distribution 
Probability matrix P(1), P(2), …, P(n)
Probability distribution over Ω P
X , and P(1), P(2), …, P(n) that satisfies the next is called Markov chain.
and P(1), P(2), …, P(n) that satisfies the next is called Markov chain.p (ω) = μω0 . μω0ω1(1) …μωn-1ωn(n)
Markov chain that does not depend on k(1≤k≤n) is called invariant Markov chain..
3-4
Invariant Markov chain P
Conditional probability P(ωs+l = (x(j) | ωl =x(i))
P(ωl = x(i))>0
P(ωs+l = (x(j) | ωl =x(i)) = p (s)ij
p (s)ij is called s class transitive probability.
3-5
Matrix P
P has a certain s0.
For arbitrary i, j p(s0)ij>0, P is called ergodic.
3-6
<Ergodic theorem>
Ergodic transitive matrix P
When Markov chain that has P is given, there exists only one probability distribution π = (π1, …, πr)that satisfies the next.
(i) πP = π
(ii) lims→∞p(s)ij = πj
4
4-1
Point x = (x1, …, xd) -∞<xi <∞
Integer 1≤i≤d
Lattice Zd
Random walk over Zd Markov chain at state space X = Zd
Random distribution over Zd p = {pz | z∈Zd}
p that satisfies the next is called to be uniform in space.
Pxy = Py-x
4-2
Locus of random walk ω = (ω0, ω1, …, ωk)
Random walk that starts from the origin ω0 = 0, pωi -ωi-1 >0
All ωs that first return to the origin toward which ω happens to be at k th Ω(k)
k>0
ω∈Ω(k)
p (ω) = pω１-ω0・・・ pωk-ωk-1
f k = ∑ω∈Ω(k) p (ω)
f 0 := 0
Random walk that satisfies the next is called to be recurrent.
∑ω∈Ω(k) f k = 1
Random walk that satisfies the next is called to be transient.
∑ω∈Ω(k) f k < 1
4-3
Arbitrary bounded sequence {an}
Generating function of {an} ∑k≥0 anzn
4-4
Generating function F(z) = ∑k≥0 f k zk P(z) = ∑k≥0 pk zk
pk = ∑ki = 0fi .pk-i
p0 = 1
F(z) = 1 – 1/ P(z)
From Abel’s theorem,
∑∞k = 1 f k = 1- lim z→1(1/ P(z) )
When ∑∞k = 0 pk = ∞ , lim z→1(1/ P(z) ) = 1/ ∑∞k = 0pk = 0
Random walk that is only ∑∞k = 0 pk = ∞ is recurrent.
4-5
e : = ∑z∈Zd zpz
Random walk that satisfies the next is called simple random walk.
(i)Unit coordinate vector e1, e2, …, ed
(ii-1)When y = ±es (1≤s≤d) , py-x = 1/2d.
(ii-2)When y ≠±es (1≤s≤d) , py-x = 0.
<Polya’s theorem>
When d = 1, 2 , simple random walk is recurrent.
When d ≥3, simple random walk is transient.
4-6
Unit vector νn = ωn / ||ωn||
Unit vector is distributed on unit sphere by being uniform in space.
4-7
From 4-1
Word : = x = (x1, …, xd) -∞<xi <∞
From 4-5
Language space : = d ≥3 and transient
From 4-6
Sentence : = νn
[References]
<On vector, sphere and Language>
<More details on Aurora Theory group>
Tokyo July 11, 2008
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