5.2.9. Let Y₁, Y₂, ... be nonnegative i.i.d. random variables with EY = 1 andP(Ym = 1) < 1. (i) Show that Xn = m≤n Ym defines a martingale. (ii) Use The-orem 5.2.9 and an argument by contradiction to show X₂ → 0 a.s. (iii) Use thestrong law of large numbers to conclude (1/n)log X₂ → c < 0.n

i.A martingale is a sequence of random variables for which the conditional expectation of the next…

.2.9. Let Y₁, Y2, ... be nonnegative i.i.d. random variables with EYm = 1 and P(Ym = 1) < 1. (i) Show that Xn = men Ym defines a martingale. (ii) Use The- rem 5.2.9 and an argument by contradiction to show X₁ → 0 a.s. (iii) Use the trong law of large numbers to conclude (1/n) log Xn → c < 0.

.2.9. Let Y₁, Y2, ... be nonnegative i.i.d. random variables with EYm = 1 and P(Ym = 1) < 1. (i) Show that Xn = men Ym defines a martingale. (ii) Use The- rem 5.2.9 and an argument by contradiction to show X₁ → 0 a.s. (iii) Use the trong law of large numbers to conclude (1/n) log Xn → c < 0.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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