How can we determine the limit of a sequence through stochastic convergence?

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SUMMARY

The discussion focuses on the relationship between a sequence H^{n} converging in probability to H and its subsequence H^{n_{k}} converging almost surely (a.s.). It establishes that performing stochastic integration with the subsequence, specifically H^{n_{k}} \cdot X, results in convergence a.s. to H \cdot X. The key question raised is how to interpret H \cdot X as the limit of H^{n} \cdot X, particularly in terms of convergence modes. The references provided include a short overview of stochastic convergence and a comprehensive MIT book on probability theory.

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  • Understanding of stochastic convergence concepts
  • Familiarity with sequences and subsequences in probability theory
  • Knowledge of stochastic integration techniques
  • Basic principles of almost sure convergence (a.s.)
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wayneckm
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Hello all,

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

Thanks very much.
 
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