Suppose that $(a_n)$ is a sequence of reals and $(e_n)$ is a sequence of iid r.v.s such that $\Pr(e_n=\pm1)=1/2$. It is well known that $\sum a_ne_n$ converges a.s. to some limit r.v. iff $\sum a_n^2 <\infty$. Almost all sample paths of this process (martingale) converge to a constant, but is it formally correct to say "the process converges to a constant"? I've seen such statements in the literature on topics other than probability and random processes and found them somewhat confusing.
For example, if a non-negative uniformly integrable martingale converges a.s. (and in the mean), then it converges to some random variable, and not to a constant, even though almost all of its sample paths converge to some constant. If a non-negative martingale is not UI, then the process may indeed converge to a constant, zero, for example. Is there anything I may be missing? Thanks for your time clarifying me on this.