.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.

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5.2.9. Let Y₁, Y₂, ... be nonnegative i.i.d. random variables with EY = 1 and
P(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 the
strong law of large numbers to conclude (1/n)log X₂ → c < 0.
n
Transcribed Image Text:5.2.9. Let Y₁, Y₂, ... be nonnegative i.i.d. random variables with EY = 1 and P(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 the strong law of large numbers to conclude (1/n)log X₂ → c < 0. n
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