Martingale Central Limit Theorem and Nonuniformly Hyperbolic ...
Central limit theorem
description
Transcript of Central limit theorem
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𝛿 𝑥𝜎
𝑃 (𝛿 𝑥 )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
𝑃 (𝑦𝑆𝑇 )
0𝑦 𝑆𝑇
𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯
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Central limit theorem
𝑃 (𝑥 )= 1𝜎 √2𝜋
𝑒− 12 ( 𝑥−𝜇𝜎 )
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HT
𝛿 𝑥𝜎
𝑃 (𝛿 𝑥 )
0 1 2 3-1-2-3
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𝛿 𝑥𝜎
𝑃 (𝛿 𝑥 )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
𝑃 (𝑦𝑆𝑇 )
0𝑦 𝑆𝑇
𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯
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Physics lab: Engineered for tightly-controlled noise
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Physics lab: Engineered for tightly-controlled noise
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Physics lab: Engineered for tightly-controlled noise
t
V
x1
x2
x3
x5
x4
Hook vibration
Uneven air flow
Thermal expansion
Laser pointer vibration
Twisting
y
𝛿 𝑦=𝛿 𝑦 (𝛿𝑥1 , 𝛿𝑥2 , 𝛿𝑥3 ,⋯ )
𝛿 𝑦≅ 𝛿 𝑦 𝐴𝑉𝐸+𝜕 (𝛿 𝑦 )𝜕 (𝛿 𝑥1 )|𝐴𝑉𝐸𝛿𝑥1+ 𝜕 (𝛿 𝑦 )
𝜕 (𝛿𝑥2 )|𝐴𝑉𝐸𝛿𝑥2+⋯𝑌
𝑋 1 𝑋 2
“Small” noise: Neglect quadratic terms in Taylor expansion
0
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𝛿 𝑥𝜎
𝑃 (𝛿 𝑥 )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
𝑃 (𝑦𝑆𝑇 )
0𝑦 𝑆𝑇
𝛿 𝑦≅𝜕 (𝛿 𝑦 )𝜕 (𝛿𝑥1 )|𝐴𝑉𝐸 𝛿𝑥1+⋯
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𝑑 𝑦𝑑𝑡
=𝜕 𝑦
𝜕𝑅+¿𝑑𝑅+¿
𝑑𝑡+ 𝜕 𝑦𝜕𝑅−
𝑑 𝑅−
𝑑𝑡¿¿
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Biology: Law of mass action and logarithms
yx2x1 x3
y
y yy
y yy y y
𝑘+¿ ¿ 𝑘−𝑅+¿¿ 𝑅−
𝑘+¿ 𝑥1𝑥2𝑥3⋯ ¿ 𝑘−𝑦+1 -1
0=𝑘+¿ 𝑥1𝑥2 𝑥3⋯−𝑘− 𝑦𝑆𝑇 ¿
𝑘−𝑦 𝑆𝑇=𝑘+¿𝑥1𝑥2 𝑥3⋯ ¿
ln ( 𝑦𝑆𝑇 )=ln ¿¿Fluctuations in x1, x2, x3, etc. are not necessarily engineered to be small. First-order Taylor-expansion might be inaccurate.
ln ( 𝑦𝑆𝑇 )=ln ¿¿ln ( 𝑦𝑆𝑇 )− ln ¿¿
𝑌 𝑋 2𝑋 1
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Biology: Law of mass action and logarithms
ln ( 𝑦𝑆𝑇 )− ln ¿¿𝑌 𝑋 2𝑋 1
A histogram of the logarithm of the concentration of y displays a normal distribution
30001507 3001500
𝑃 [ln ( 𝑦𝑆𝑇 ) ]
5 6 7 82 3 4ln ( 𝑦𝑆𝑇 )
𝑃 (𝑦𝑆𝑇 )
200 3001000 𝑦 𝑆𝑇
yST = e4 = 55
e5 = 150
e6 = 400