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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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up vote 3 down vote accepted

"Constant" in the context of probability usually implies "non-random", so I would say it is not appropriate to use the word "constant" in this case.

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.

More generally, calling something a "constant" implies that its value does not depend on some other parameter, which is usually understood from context. However, this certainly can be a source of confusion when it isn't clear precisely what depends on what. If there is any doubt, it is usually best to point out exactly what your "constant" does and does not depend on.

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This seems to me like a definitive answer. – Did Oct 14 '11 at 5:54

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