Donsker Theorem and its application Vadym Omelchenko
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- Slide 1
- Vadym Omelchenko
- Slide 2
- Definition
- Slide 3
- Donsker Theorem
- Slide 4
- Proof
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Proof of the tightness
- Slide 10
- Proof (Proof of the Lemma)
- Slide 11
- Slide 12
- Hence both (A) and (B) imply (3) which is the affirmation of the theorem. QED
- Slide 13
- Proof
- Slide 14
- Slide 15
- Having proved the assertion of this theorem for finite- dimensional distributions and having proved the tightness we have proved the theorem. QED
- Slide 16
- Application of Donsker Theorem
- Slide 17
- Unit Dimension {-1,+1} N=20 N=60 N=1000
- Slide 18
- Application of Donskers Theorem More important than this qualitative interpretation is the use of Donsker's theorem to prove limit theorems for various functions of the partial sums
- Slide 19
- Application of Donskers Theorem
- Slide 20
- Random Walk and Reflection Principle
- Slide 21
- Hence we have:
- Slide 22
- Combining the results (**) and (***) we have:
- Slide 23
- Slide 24
- Functions of Brownian M. Paths
- Slide 25
- Slide 26
- The Arc Sine Law
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Example(1) Normal and Student-t
- Slide 33
- Example (2)
- Slide 34
- Brownian Bridge
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