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