What are the recent works and research scope in asymptotic inference (large sample theory)? What are some current significant theoretical work that has been done in the field of asymptotic inference / large sample theory? What is the research scope in this field right now? Is there any open problem or specific areas where the theory is developing in recent times? Or is it a dead subject with no scope of further development?
I'd be grateful if anyone can answer my questions or give any source/reference where I can search.
 A: I am probably less up-to-date than you in this field, so rather than giving you some fish, I am going to try to teach you to fish.  I also hope that  this answer might be more broadly interesting to readers that also want to look up statistical literature, but are interested in a different topic than you.  Please forgive me if any of this is well-known to you; it is not intended to be condescending, but merely to give some general advice that might be useful to many readers of this site.
Your question is essentially asking for a recent literature review of a field of interest to you, where you have some partial familiarity with the subject.  There are a lot of resources you can use to give you suggestions on conducting a literature review, and in fact, there are also a few book sections on the topic (see e.g., O'Leary 2004, Jesson 2011).  Since we live in the internet-age, much of this is a matter of becoming skilled at using search techniques to identify useful literature.  If you are at a university then you probably have access to the Web of Science portal, where you can search for literature via keywords, and also analyse the results by year of publication and other variables.  If you do not have access to this then you can also use Google Scholar, which also has substantial search facilities.  (Google-Scholar has a broad search net, including academic articles, books, conference proceeding and pre-prints, and it also auto-updates citation metrics.  The wide scope of this search engine is both a blessing and a curse depending on context.)
Finding important literature in a desired field of study is really just a matter of learning good search techniques and then having a lot of tenacity.  Initial search results lead to more citations, which lead to more results, which lead to more citations, virtually ad infinitum.  Once you have extended your search widely, you will usually be able to find the items that come up again and again in searches, and this will usually give you a reasonable idea of the most "significant" works.

An example of searching for your literature of interest: Here are some steps you could take to find what you're looking for through Google-Scholar:


*

*Read up on how to do advanced Google-Scholar search queries;

*Start with searches using basic keywords you expect to see in that field.  For example, for your query, I would start with "statistics asymptotic theory", and maybe also search with a restriction to works published since 2014.  Note that some works will be republished books that were initially published prior to the date restriction, but these can easily be identified by clicking on the tab that says X related versions.

*Go through the pages of search results and pull out the ones that look like they fall within the field you are interested in.  If you only want to look at "significant" works, this is usually identifiable prima facie by looking at the number of citations relative to age.  The most highly-cited works should show up near the top of your search results, and these are the most "significant" works, in the sense of being cited most often.

*Read some of the identified papers/books and check their citations for more leads to other papers.  You can also go the other way and use Google-Scholar to get a list of all the publications that this one was cited by.  (This latter technique is usually a bit less useful, because a lot of papers cite things you are looking at, without being focused on the same subject area of interest.)

*Sometimes you get especially lucky and you find that there has been a recent published literature review of the field you are interested in.  For example, on the second page of my search results, I find that Gomes and Giullou (2015) is a review of literature and results in extreme value theory, with a healthy emphasis on asymptotics.  One more Google search finds me an accessible pdf version and now I have a whole paper reviewing the subject, with another 258 citations!  (Perhaps this is not quite what you're looking for?)

*Continue this game of whack-a-mole until you find what you need or pass out from exhaustion.  Every new paper you find leads to a new list of citations, and every new citation leads to a new paper!
A: I would point out that "Asymptotics/Limit Theory" is the general term covering all cases where we study Approximation theory, while the "sample size goes to infinity Asymptotics" is just a particular subfield in there. 
Viewing the field as a user of its results, I would not say that major things and breakthroughs are happening for some time now (of the variety that will spill over to Statistics etc).  
What one could see as a largely open direction, is Limiting theory for non-stationary and non-ergodic processes, since so much non-stationarity and non-ergodicity exists in the real world.
Anirban DasGupta's book "Asymptotic Theory of Statistics and Probability" (2008) is perhaps the best panorama of the field.
