Correct answer: (A) P({X_n>(ε/2)ln n} i.o.)=1 for 0<ε≤1.; (B) P({X_n<(ε/2)ln n} i.o.)=1 for 0<ε≤1.; (D) P({X_n<(ε/2)ln n} i.o.)=1 for ε>1.
Explanation
The upper-event probability is n^(−ε). Its series diverges for ε≤1 and converges for ε>1;independence and the two Borel–Cantelli lemmas prove A and refute C. Lower-event probabilities are 1−n^(−ε);their series diverges for every ε>0 and they are independent,proving B,D. **A,B,D**.
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