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4. a) Let Z:= supo< 0. Hint: Kolmogorov's inequality. b) Show that lim t. X₁ = 0 P-a.s. t-x for every a > 1/2. Hint: Use a), Aufgabe 3, and the strong law of large numbers. c) Conclude that Y= (Yt) te[0,7] with Y:= t X₁/t for t> 0 and Yo:= 0 is a Brownian motion.

4. a) Let Z:= supo< 0. Hint: Kolmogorov's inequality. b) Show that lim t. X₁ = 0 P-a.s. t-x for every a > 1/2. Hint: Use a), Aufgabe 3, and the strong law of large numbers. c) Conclude that Y= (Yt) te[0,7] with Y:= t X₁/t for t> 0 and Yo:= 0 is a Brownian motion.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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4. a) Let Z suPo<t<TX. Show that Z is A-B-measurable and P({Z ≥ u}) <T/u² for every u> 0. Hint: Kolmogorov's inequality. b) Show that lim ta X₂ = 0 t-x P-a.s. for every a > 1/2. Hint: Use a), Aufgabe 3, and the strong law of large numbers. c) Conclude that Y = (Yt)te[0,7] with Y=t X1/t for t> 0 and Yo:= 0 is a Brownian motion.
Transcribed Image Text:4. a) Let Z suPo<t<TX. Show that Z is A-B-measurable and P({Z ≥ u}) <T/u² for every u> 0. Hint: Kolmogorov's inequality. b) Show that lim ta X₂ = 0 t-x P-a.s. for every a > 1/2. Hint: Use a), Aufgabe 3, and the strong law of large numbers. c) Conclude that Y = (Yt)te[0,7] with Y=t X1/t for t> 0 and Yo:= 0 is a Brownian motion.
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