Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 63, Number 2 (2019), 219-257.
Polynomial time relatively computable triangular arrays for almost sure convergence
We start from a discrete random variable, , defined on and taking on values with equal probability—any member of a certain family whose simplest member is the Rademacher random variable (with domain ), whose constant value on is . We create (via left-shifts) independent copies, , of and let . We let be the quantile of . If is Rademacher, the sequence is the equiprobable random walk on with domain . In the general case, follows a multinomial distribution and as varies over the family, the resulting family of multinomial distributions is sufficiently rich to capture the full generality of situations where the Central Limit Theorem applies.
The provide a representation of that is strong in that their sum is equal to pointwise. They represent only in distribution. Are there strong representations of ? We establish the affirmative answer, and our proof gives a canonical bijection between, on the one hand, the set of all strong representations with the additional property of being trim and, on the other hand, the set of permutations, , of , with the property that we call admissibility. Passing to sequences, , of admissible permutations, these provide a complete classification of trim, strong triangular array representations of the sequence . We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function which embodies the complexity of itself. The trim, strong triangular array representation corresponding to the second of these is as close as possible to the representation of provided by the .
Illinois J. Math., Volume 63, Number 2 (2019), 219-257.
Received: 29 December 2016
Revised: 3 April 2019
First available in Project Euclid: 1 August 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F15: Strong theorems 60F05: Central limit and other weak theorems 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19] 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15] 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Dobrić†, Vladimir; Garmirian, Patricia; Skyers, Marina; Stanley, Lee J. Polynomial time relatively computable triangular arrays for almost sure convergence. Illinois J. Math. 63 (2019), no. 2, 219--257. doi:10.1215/00192082-7768719. https://projecteuclid.org/euclid.ijm/1564646432