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Vadym Omelchenko

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Definition

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Donsker Theorem

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Proof

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Proof of the tightness

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Proof (Proof of the Lemma)

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

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Proof

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

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Application of Donsker Theorem

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

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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

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Application of Donskers Theorem

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Random Walk and Reflection Principle

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Hence we have:

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

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Functions of Brownian M. Paths

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The Arc Sine Law

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Example(1) Normal and Student-t

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Example (2)

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Brownian Bridge