Higher-order dangers and precisely constructed taxa in models of randomness.

Abstract

The certification, construction, and delineation of individual, infinite-length "random" sequences have been longstanding yet incompletely resolved problems. We address this topic via the study of normal numbers, which often have been viewed as reasonable proxies for randomness, given their limiting equidistribution of subblocks of all lengths. However, limitations arise within this perspective. First, we explicitly construct a normal number that satisfies the law of the iterated logarithm yet exhibits pairwise bias toward repeated values, rendering it inappropriate for any collection of random numbers. Accordingly, we deduce that the evaluation of higher-order block dynamics, even beyond limiting equidistribution and fluctuational typicality, is imperative in proper evaluation of sequential "randomness." Second, we develop several criteria motivated by classical theorems for symmetric random walks, which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums, including rates of sign changes, patterns of return times to 0, and the extent of fairness of the sequence. Such characteristics generally are unaddressed in most evaluations of randomness. More broadly, we can differentiate normal numbers both on the basis of multiple distinct qualitative attributes and quantitatively via a spectrum of rates within each attribute. Furthermore, we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications, including profound biases. Overall, we elucidate the vast diversity within the category of normal sequences.