In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear models) and the statistic's distribution is derived using concepts/properties of Brownian motion, Functional CLT, etc. See this for example, and this question for my motivation.
Since I do not have a strong background in Stochastic Processes, I am looking for resources (hopefully a book) which covers relevant topics from stochastic processes and asymptotics so that I can make sense of the derivations in papers on unit root tests.
I tried to check in books on Stochastic Processes but their coverage is wider than what I am looking for. I did found two interesting links: this and this, which have a rather focused use of stochastic process concepts for unit roots. I am looking for similar resources - preferably a book.