Effective convergence is the study of weak convergence of measures and vague convergence within the framework of computable analysis (Type-2 computability theory). The theory, developed by McNicholl-Rojas (2021) and extended by Rojas (2021), asks when classical convergence notions can be “effectivised” — that is, when the convergence is witnessed by computable moduli and produces computable limits.
A sequence {mu_n} in M(R) is uniformly computable if each mu_n is a computable measure, uniformly in n. For such sequences:
Effective weak convergence: mu_n effectively weakly converges to mu if for every computable f in C_b(R), the integrals converge and one can compute a modulus of convergence from an index of f and a bound on |f|.
Effective vague convergence: mu_n effectively vaguely converges to mu if the same holds for computable f in C_K(R) (compactly supported), with a modulus computable from an index of f and a name of supp(f).
Key results:
- Effective weak convergence = effective Prokhorov convergence (Theorem 4.1)
- Uniform and non-uniform definitions of effective vague convergence are equivalent (Theorem 5.3)
- Effective weak convergence always produces a computable limit
- Effective vague convergence may produce an incomputable limit, even when finite (Proposition 5.6)
- Effective vague + computable modulus for total masses ⇒ effective weak (Theorem 5.8)
- For computable probability measures: effective vague = effective weak (Corollary 5.12)
The striking result is Proposition 5.6: there exists a uniformly computable sequence that effectively vaguely converges to a finite but incomputable measure. This has no classical analogue — classically, the limit of any convergent sequence of finite measures is a finite measure. The “vagueness” in effective vague convergence manifests as potential incomputability of limits.
Key Details
- Framework: Type-2 computability / TTE, measures represented by their actions on rational open intervals
- Key distinction: effective weak convergence preserves computability; effective vague convergence does not
- Bridge condition: computability of mu(R) is necessary and sufficient for effective vague limits to be computable
- Classical recovery: for probability measures, effective weak = effective vague, recovering the classical theorem
- Effective Portmanteau theorem: five equivalent characterisations of effective weak convergence (Theorem 3.5)