Let {X„}nen be a symmetric random walk (X„ = ±1 w.p. 1/2) on (-00, 1]NZ. Set S, = E- Xk.a) Show that for all 0 € RM = (sech 0)"e0S.is a martingale, where sech is the hyperbolic secant.b) Considerwhere T = inf{n : S, = 1}. Letting n → ∞, and then 0 → 0 show that P(7 < x) = 1.

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Let {X,}neN be a symmetric random walk (X„ = ±1 w.p. 1/2) on (-0o, 1]nZ. Set S, = E X- a) Show that for all 0 €R %3D M = (sech 0)"e"S. is a martingale, where sech is the hyperbolic secant. b) Consider where 7 = inf{n : S, 1}. Letting n → 0, and then 0→ 0 show that P(7 < 0) = 1.

Let {X,}neN be a symmetric random walk (X„ = ±1 w.p. 1/2) on (-0o, 1]nZ. Set S, = E X- a) Show that for all 0 €R %3D M = (sech 0)"e"S. is a martingale, where sech is the hyperbolic secant. b) Consider where 7 = inf{n : S, 1}. Letting n → 0, and then 0→ 0 show that P(7 < 0) = 1.

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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Let {X„}nen be a symmetric random walk (X„ = ±1 w.p. 1/2) on (-00, 1]NZ. Set S, = E- Xk.
a) Show that for all 0 € R
M = (sech 0)"e0S.
is a martingale, where sech is the hyperbolic secant.
b) Consider
where T = inf{n : S, = 1}. Letting n → ∞, and then 0 → 0 show that P(7 < x) = 1.

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