My (naive) intuition of an information source is that it's just a "black box" that emits symbols ranging over a finite alphabet. What's missing from this definition?
On the wikipedia page for an information source, it says:
In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution.
I have bolded the two parts that I'm confused about.
First, the wikipedia page for stationary distribution isn't very helpful since it's a disambiguation page and I'm not sure which link is relevant to the formal definition of an information source.
And second, my conception of a "random variable" was that it maps events (of a probability space) to symbols. Is the term "random variables" appropriate here? Wouldn't it make more sense to say "an information source is a sequence of symbols ranging over a finite alphabet"? By using the term "random variables" the Wikipedia definition implies that an information source is a sequence of functions.
If someone could correct my understanding here that'd be great!
Wiki doesn't say something wrong about the definition of an information source. Your own definition is also not wrong but probably incomplete.
Random variable is a function defined from a sample space to a measurable space. If you have a collection of random variables, they do create random samples, i.e. a random vector as in your definition. whenever you want to obtain another vector you can get it from the collection of random variables you have. Stationary means that the distributions of the random variables are fixed and do not change in time.
A set of random variables actually does not emit a sequence of functions, it emits only sequence of symbols. The source has the capability of outputing any kind of sequence which can in fact be possible. For this, one needs a set of random variables.