A constructive law of large numbers with application to countable Markov chains

Let X1,X2, . . . be a sequence of identically distributed, pairwise independent random variables with distribution P. Let the expected value be μ < ∞. Let S n = ∑i=1 Xi. It is well-known that S n/n converges to μ almost surely. We show that this convergence is effective in (P,μ). In particular, if P,μ are computable then the convergence is effective. On the other hand, if the convergence is effective in P then μ is computable from P. The effectiveness of convergence is detached in the sense that nothing can be inferred about the speed of convergence in the law of large numbers from the speed of convergence in computing P and μ. This theorem can be used to show an effective renewal theorem, which then can be used to prove an effective ergodic theorem for countable Markov chains. The last result is a special case of effective ergodic theorems proven by Avigad-Gerhardy-Towsner and Galatolo-Hoyrup-Rojas, but we hope that the direct constructivization of the probability-theory proofs is still useful.