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There is always a confusing question in my mind regarding sequence and subsequence, particularly in the field of probability theory and stochastic integration.

Given a sequence [tex]H^{n}[/tex] which converges in probability to [tex]H[/tex], we know that there exists a subsequence [tex]H^{n_{k}}[/tex] converging a.s., suppose now we perform some sort of stochastic integration by using this subsequence, [tex]H^{n_{k}} \cdot X[/tex], and this converges a.s. to [tex]H \cdot X[/tex], so how can we conclude this `limit' [tex]H \cdot X[/tex] with the original sequence [tex]H^{n}[/tex], i.e. is [tex]H \cdot X[/tex] in what sense the limit of [tex]H^{n} \cdot X[/tex]? a.s.? some other modes? or no conclusion?

Thanks very much.

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# Sequence and Subsequence

Can you offer guidance or do you also need help?

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