Question: 1.13. Let {Xn} be a sequence of identically distributed strictly positive random variables. For any cp such that cp(n )/n ---+ 0 as n ---+

1.13. Let {Xn} be a sequence of identically distributed strictly positive random variables. For any cp such that cp(n )/n ---+ 0 as n ---+ 00, show that JO{SII > cp(n) i.o.} = 1, and so Sn ---+ 00 a.e. [HINT: Let Nn denote the number of k ~ 11 such that Xk ~ cp(n)ln. Use Chebyshev'S inequality to estimate 9{Nn > n12} and so conclude 9{Sn > cp(n)/2} ~ 1 -2F(cp(n)ln). This pro blem was proposed as a teaser and the rather unexpected solution was given by Kesten.]

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