4. Let X₁, i = 1,2,..., be independent random variables with P(X; = 1) = ½, P(X; = − 1) = 1. Put Fo= {2,0}, F₁ = 0(X₁, X₂,..., X), n ≥ 1. Define So = 0, Sn = ΣX₁, n ≥ 1. Prove that for any X > 0, eAsn in 20
4. Let X₁, i = 1,2,..., be independent random variables with P(X; = 1) = ½, P(X; = − 1) = 1. Put Fo= {2,0}, F₁ = 0(X₁, X₂,..., X), n ≥ 1. Define So = 0, Sn = ΣX₁, n ≥ 1. Prove that for any X > 0, eAsn in 20
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:4. Let X₁, i = 1,2,..., be independent random variables with P(X₂ = 1) = P(X; = -1) = 1/2. Put
Fo= {2,0}, F₁ =0(X₁, X2, ..., Xn), n ≥ 1.
Define So= 0, S₁ = ΣX₁, n ≥ 1.
Prove that for any À > 0,
exSn
(cosh(A))n ¹
is a martingale with respect to {F, n ≥ 0}, where
Yn
in 20
1
cosh(A) =(e^ + e^^).
We can use the conclusion that:
Zn = S2-n, n ≥ 0 is a martingale with respect to {Fn, n >0}.
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