4. Let X₁, i = 1,2,..., be independent random variables with P(X₂ = 1) = P(X; = -1) = 1/2. PutFo= {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 ¹is a martingale with respect to {F, n ≥ 0}, whereYnin 201cosh(A) =(e^ + e^^).We can use the conclusion that:Zn = S2-n, n ≥ 0 is a martingale with respect to {Fn, n >0}.

From the given information, Define,

Answered: 4. Let X₁, i = 1,2,..., be independent…

A First Course in Probability (10th Edition)

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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