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 which converges in probability to , we know that there exists a subsequence converging a.s., suppose now we perform some sort of stochastic integration by using this subsequence, , and this converges a.s. to , so how can we conclude this `limit' with the original sequence , i.e. is in what sense the limit of ? a.s.? some other modes? or no conclusion?
Thanks very much.