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Donsker Theorem and its application

Donsker Theorem and its application

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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