What sort of "alternative" probability theories are out there in which the methods of proof are inherently constructive?

I know of a number of theorems that say that if you take an infinite sequence of i.i.d. random variables of thus-and-such a kind (let's say that they're fair bits, for definiteness), and use them in a specified fashion to generate a sequence of combinatorial objects of a particular sort, and rescale those combinatorial objects in a time-dependent fashion, then the rescaled objects converge to some sort of limit object with probability 1. However, the proofs that I know are ineffective, in the sense that the proofs don't give you a way to construct any particular infinite sequence of bits such that, if you use them as described above, the convergence occurs.

Well, sort of. In each case of this situation occurring, there's a way to "cheat" by using the theorem itself to guide the choice of bits; you can just choose your bits to have the behavior that you're trying to prove. Is there some principled way to rule out such "cheating"? When it comes to cheating, I believe that "I know it when I see it", but I don't know how to formulate a precise definition of cheating that captures my intuitions.

A web search turned up a talk on "Applications of Effective Probability Theory to Martin-Lof Randomness" (http://www.loria.fr/~hoyrup/icalp_slides.pdf), which is one example of the kind of theory I mean. Are there others?