# If almost all sample paths of a process converge to a constant, is it okay to say the process itself converges to a constant?

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.

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Let's make the situation explicit. We have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a sequence of random variables $(e_n)$, which are measurable maps from $\Omega$ to $\mathbb{R}$. Under the assumption that $\sum a_n^2 < \infty$, we know that $\sum a_n e_n(\omega)$ converges for $\mathbb{P}$-almost every $\omega \in \Omega$. But the limit, $L(\omega)$, depends on $\omega$. For a given $\omega$, $L(\omega)$ is some number, but we shouldn't call it a constant, because it varies with $\omega$.
If there were a single, fixed real number $c$ such that we had $L(\omega) = c$ for ($\mathbb{P}$-almost) every $\omega \in \Omega$, then it would be reasonable to call $L$ a constant. That isn't the case here.