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# Vadym Omelchenko. Definition Donsker Theorem Proof.

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

Vadym Omelchenko

Definition

Donsker Theorem

Proof

Proof of the tightness

Proof (Proof of the Lemma)

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

Proof

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

Application of Donsker Theorem

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

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

Application of Donsker’s Theorem

Random Walk and Reflection Principle

Hence we have:

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

Functions of Brownian M. Paths

The Arc Sine Law

Example(1) Normal and Student-t

Example (2)

Brownian Bridge

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