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∗

∗

M(x)≡X

p:U(p)=x

2−p|

p|p|

U x

x

M(xt|x1, . . . , xt−1)→1

t→ ∞ xii

3·109·2/8/10242= 715

1.5

715 ·10% ·1.5% ·1% =

1.0

U

U pi

L

t xi

Kt(x) = min

p{|p|+ log t:U(p) = x t }

(xi, po

i)

po

i={p:Kt(xi) = |p|+ log t, |p| ≤ L}

po

ixip

ri(L) = |xi|−|p|

|xi|−|po

i|∈[0,1]

x{xi}

R

L

R(L) = hri(L)i

L R 1L

L

x

p|p| ≤ L

O(2n−m)

n m

Θ(2n−m)

π3,1,4,1,5,9,2,6,5,3, . . .

π

1,3,1,3,2,4,2,4,2,3,5,3,5,3,5,4,6,4,6,4,6,4,

1,4,1,4,2,5,2,5,2,3,6,3,6,3,6,4,7,4,7,4,7,4, ...

1,3, . . .

2

1,3,5,7,9, . . .

n

{−1,−2,...,−n}n

n

0001111100000000

{0,1}p= 0.5

H0=−16 log2(p) = 16

n l n l

H= log216 = 4

H

1−H+H

H0

H(1) = 2 ·10 ·log2(1024) =

200

1−200/1024 = 80.4%

ni= 100 ·i li= 4

100 H(2) = 20

1−20/1024 = 98%

π

{1,...,14}

α

8 8

0.5

0.46878

5 101 46878

1011011100011110

0.5 0.46878

1

/2,1

/3,1

/4

1/π

l

l1, l2, l3, . . .

l≤l1∧l > l2∧l≤(l3−l2)2/l1

lr

l≥lr

l=lr± 1

a b 1 1000 99.9%

a=b

lrl

l≥l0l

l0a≤l0≤b

X X = 1

H1:l0=lrH2:l0

H1

l p(X= 1|l, H1) = Θ(l−lr) Θ

H2

a b p(X= 1|l, H2) = l−a

b−a

p(X= 1|l) = p(X= 1|l, H1)p(H1) + p(X= 1|l, H2)p(H2) =

Θ(l−lr)β+l−a

b−a(1 −β) = 1

2

β=p(H1)=1−p(H2)

1

2l

H1β.1lr

l0

1

2

l=lr

n×n

n2

2 log2n m n2

1+2mlog2nn2

l

1+2llog2n1+2mlog2n

l= 12 n= 128

H= 1 + 2llog2n= 169

H=n2= 16384

⊂ ⊂ ⊂

⊂ ⊂ ⊂ ⊂ ⊂

n n(n−1)

n(n−1)