I am new to the world of stochastic processes and Brownian motion, and am having a hard time finding easily digestible reference material (especially ones with well-worked out examples). So, in a way this question is more of a request for sources/references than a specific answer.
To give a little context to where I am approaching this from, it is specifically from the class of statistics referred to as "group sequential testing" (though there are several similar or analagous names, such as "interm analyses", "conditional power calculations", "futility analyses", etc.). I am posting this here rather than on CrossValidated simply because cursory searches reveal that people here are more familiar with the basics of stochastic processes than seems to be the case over on the stats-specific StackExchange, so I feel that I will get more helpful advice/answers here.
Anyway, the primary textbook for this field of statistical analyses is Statistical Monitoring of Clinical Trials: A Unified Approach, by Proschan, Lan, and Wittes. The basic mathematical framework for much of the book is Brownian motion; that is, the theory behind many of the group sequential tests outlined in the book is founded on the fact that standardized test statistics evaluated over time can be seen to satisfy the properties of Brownian motion. What I am trying to do is prove whether or not certain classes of non-parametric statistics (specifically the Wilcoxon rank-sum) satisfy these properties, and thus allow one to use this broader framework in that context.
So, what I am looking for are sources or references with examples of demonstrating how some arbitrary process can be shown to satisfy the properties of Brownian motion. When I search this site or Google, I can find dozens of examples, but none of them are quite that helpful, since the are generally all based on proving that some function of a Weiner process is itself a Weiner process (e.g. this question or this one). These aren't terribly illustrative to me, since the proofs all rely on the fact that some element of the process under consideration is already given to satisfy the properties of Brownian motion, rather than "starting from scratch", so to speak.
Now, I do know the basic steps for proving a process is Brownian motion (this question has them delineated rather nicely). However, I am having a difficult time finding any worked examples of these steps being applied to a process "from scratch" (which is especially critical since I am unfamiliar with a lot of the notations used in stochastic processes, which are often subtly different than those taught to us statisticians).
Can anybody point me in the direction of some nice examples or references that would be helpful, here?