Vanishing proportion under sublinear stochasticity parameters

Determine whether N(k,α,β)/2^k tends to zero whenever k tends to infinity while α/k and β/k both tend to zero, where N(k,α,β) denotes the number of (α,β)-stochastic finite objects of complexity at most k.

Background

This is the specific convergence question included among Kolmogorov’s problems. It asks whether the proportion of objects classified as (α,β)-stochastic becomes negligible when both the model-complexity allowance α and the deficiency allowance β are sublinear in the object-complexity bound k. As with the preceding question, the paper cautions that N may have been intended to count nonstochastic objects rather than stochastic ones, so the precise intended formulation remains subject to that historical definitional ambiguity.