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

Thanks very much.