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# Donsker Theorem and its application

Download Presentation ## Donsker Theorem and its application

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1. DonskerTheorem and its application VadymOmelchenko

2. Definition

3. Donsker Theorem

4. Proof

5. Proof

6. Proof

7. Proof

8. Proof

9. Proof of the tightness

10. Proof (Proof of the Lemma)

11. Proof (Proof of the Lemma)

12. Proof (Proof of the Lemma) • Hence both (A) and (B) imply (3) which is the affirmation of the theorem. QED

13. Proof

14. Proof

15. Proof • Having proved the assertion of this theorem for finite-dimensional distributions and having proved the tightness we have proved the theorem. QED

16. Application of Donsker Theorem

17. Unit Dimension {-1,+1} • N=20 N=60 • N=1000

18. 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 sums

19. Application of Donsker’s Theorem

20. Random Walk and Reflection Principle

21. Hence we have:

22. Combining the results (**) and (***) we have:

23. Functions of Brownian M. Paths

24. Functions of Brownian M. Paths

25. The Arc Sine Law

26. The Arc Sine Law

27. The Arc Sine Law

28. The Arc Sine Law

29. The Arc Sine Law

30. The Arc Sine Law

31. Example(1) Normal and Student-t

32. Example (2)

33. Brownian Bridge