(a) Generalize the Borel-Cantelli lemmas- Suppose Xn are nonnegative I.docx

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(a) Generalize the Borel-Cantelli lemmas: Suppose X n are nonnegative If E [ X n ] < , then X n converges with probability 1 . If the X n are in depen dent and uniformly bounded, and if E [ X n ] = , then X n diverges witt probability 1. (b) Construct independent, nonnegative X n such that X n converges with probability 1 but E [ X n ] diverges For an extreme example, arrange that P [ X n > 0 i.o. ] = 0 but E [ X n ] . .

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