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.
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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