And stay there? Or visit it as part of, for instance, a random walk?
And stay there, except for occasional digressions.
In other words, assuming I understand the claim: as time approaches infinity, so the probability of a randomly selected country being democratic approaches 1.
I often hear people speak of democracy as the next, or the final, inevitable stage of human social development. Its inevitability is usually justified not by describing power relations that result in democracy being a stable attractor, but in terms of morality - democracy is more "enlightened". I don't see any inevitability to it - China and the Soviet Union manage(d) to maintain large, technologically-advanced nations for a long time without it - but suppose, for the sake of argument, that democracy is the inevitable next stage of human progress.
The May 18 2012 issue of Science has an article on p. 844, "Ancestral hierarchy and conflict", by Christopher Boehm, which, among other things, describes the changes over time of equality among male hominids. If we add its timeline to recent human history, then here is the history of democracy over time in the evolutionary line leading to humans:
There are two points to observe in this data:
I do believe "progress" is a meaningful term. But there isn't some cosmic niceness built into the universe that makes everything improve monotonically along every dimension at once.