1 / 34

Donsker Theorem and its application

Donsker 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). Proof (Proof of the Lemma).

gates
Download Presentation

Donsker Theorem and its application

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related