Quenched convergence for annealed inclusion-induced randomizations

Establish quenched convergence to a nondegenerate Brownian motion for the inclusion-induced random-walk randomizations whose annealed convergence is proved when the random coefficients are grouped either by the pair consisting of time level and subset cardinality, or by subset cardinality alone, under the respective elliptic Bernoulli-parameter assumptions stated in the paper.

Background

The paper studies random bootstrap transformations of a symmetric simple random walk, represented through random Bernoulli coefficients in either the totally unordered or inclusion-induced basis. It distinguishes annealed convergence, which averages over both the walk and the random coefficients, from quenched convergence, which conditions on a realization of the coefficient environment.

For the inclusion-induced representation, the paper proves annealed Brownian convergence in two settings: when independent blocks are indexed by the time level and the subset cardinality, with parameters depending on the level and cardinality; and when independent blocks are indexed by subset cardinality, with parameters depending only on cardinality. The corresponding quenched convergence results are explicitly identified as unresolved.