Suppose we have a sequence of positive random variables
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Answered question
2022-05-29
Suppose we have a sequence of positive random variables . I am trying to prove a characterization of almost sure convergence. It states that almost surely iff for every , and . If I assume almost sure convergence, then the implication is easy but I am not being able to prove the other way round.
Answer & Explanation
concludirgt
Beginner2022-05-30Added 11 answers
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Since all the random variables are positive,
Thus,
But
That is, is equivalent to
and, similarly, is equivalent to
The last condition is equivalent to .
Trevor Wood
Beginner2022-05-31Added 5 answers
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Let