Theorem and its application Vadym Omelchenko Definition Donsker Theorem Proof Proof Proof Proof Proof Proof of the tightness Proof Proof of the Lemma Proof Proof of the Lemma ID: 421819
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Donsker Theorem and its application
Vadym
OmelchenkoSlide2
DefinitionSlide3
Donsker TheoremSlide4
ProofSlide5
ProofSlide6
ProofSlide7
ProofSlide8
ProofSlide9
Proof of the tightnessSlide10
Proof (Proof of the Lemma)Slide11
Proof (Proof of the Lemma)Slide12
Proof (Proof of the Lemma)
Hence both (A) and (B) imply (3) which is the affirmation of the theorem. QEDSlide13
Proof
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Proof
Slide15
Proof
Having proved the assertion of this theorem for finite-dimensional distributions and having proved the tightness we have proved the theorem. QEDSlide16
Application of Donsker TheoremSlide17
Unit Dimension {-1,+1}
N=20 N=60
N=1000Slide18
Application of Donsker’s Theorem
More important than this qualitative interpretation is the use of
Donsker's
theorem to prove limit theorems for various functions of the partial sumsSlide19
Application of Donsker’s TheoremSlide20
Random Walk and Reflection Principle
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Hence we have:Slide22
Combining the results (**) and (***) we have:Slide23Slide24
Functions of Brownian M. PathsSlide25
Functions of Brownian M. PathsSlide26
The Arc Sine LawSlide27
The Arc Sine LawSlide28
The Arc Sine LawSlide29
The Arc Sine LawSlide30
The Arc Sine LawSlide31
The Arc Sine LawSlide32
Example(1) Normal and Student-tSlide33
Example (2) Slide34
Brownian Bridge