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

In summary, the conversation discusses the confusion between sequence and subsequence in the field of probability theory and stochastic integration. It is stated that a sequence H^{n} converges in probability to H, and a subsequence H^{n_{k}} converges almost surely. The question then arises about how to conclude the limit H \cdot X with the original sequence H^{n} and in what sense H \cdot X is the limit of H^{n} \cdot X. The provided resources offer a short and long version of information on stochastic convergence for further understanding.
  • #1
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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1. What is a sequence?

A sequence is a specific order in which objects, events, or numbers follow each other. In science, sequences are often used to describe the order of nucleotides in DNA or the order of amino acids in a protein.

2. How is a subsequence different from a sequence?

A subsequence is a part of a larger sequence that follows the same order. It may contain some, but not all, of the elements in the original sequence. A sequence, on the other hand, includes all of the elements in a specific order.

3. What is the importance of studying sequences and subsequences?

Sequences and subsequences are important in understanding the structure and function of biological molecules like DNA and proteins. They can also provide information about the evolutionary relationships between different organisms.

4. How are sequences and subsequences used in research?

Scientists use sequences and subsequences to identify and compare genetic material between different organisms. This can help in understanding the genetic basis of diseases and in developing treatments or cures.

5. Can sequences and subsequences change over time?

Yes, sequences and subsequences can change over time through mutations or other genetic processes. This is an important factor in the evolution of species and the development of genetic diseases.

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