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Vadym Omelchenko. Definition Donsker Theorem Proof.

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Presentation on theme: "Vadym Omelchenko. Definition Donsker Theorem Proof."— Presentation transcript:

1 Vadym Omelchenko

2 Definition

3 Donsker Theorem

4 Proof

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9 Proof of the tightness

10 Proof (Proof of the Lemma)

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

13 Proof

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15 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

24 Functions of Brownian M. Paths

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26 The Arc Sine Law

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32 Example(1) Normal and Student-t

33 Example (2)

34 Brownian Bridge


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