## Illinois Journal of Mathematics

### Polynomial time relatively computable triangular arrays for almost sure convergence

#### Abstract

We start from a discrete random variable, $\mathbf{O}$, defined on $(0,1)$ and taking on $2^{M+1}$ values with equal probability—any member of a certain family whose simplest member is the Rademacher random variable (with domain $(0,1)$), whose constant value on $(0,1/2)$ is $-1$. We create (via left-shifts) independent copies, $\mathbf{X}_{i}$, of $\mathbf{O}$ and let $\mathbf{S}_{n}:=\sum _{i=1}^{n}X_{i}$. We let $\mathbf{S}^{*}_{n}$ be the quantile of $\mathbf{S}_{n}$. If $\mathbf{O}$ is Rademacher, the sequence $\{\mathbf{S}_{n}\}$ is the equiprobable random walk on $\mathbb{Z}$ with domain $(0,1)$. In the general case, $\mathbf{S}_{n}$ follows a multinomial distribution and as $\mathbf{O}$ 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 $\mathbf{X}_{1},\ldots ,\mathbf{X}_{n}$ provide a representation of $\mathbf{S}_{n}$ that is strong in that their sum is equal to $\mathbf{S}_{n}$ pointwise. They represent $\mathbf{S}^{*}_{n}$ only in distribution. Are there strong representations of $\mathbf{S}^{*}_{n}$? 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, $\pi _{n}$, of $\{0,\ldots,2^{n(M+1)}-1\}$, with the property that we call admissibility. Passing to sequences, $\{\pi _{n}\}$, of admissible permutations, these provide a complete classification of trim, strong triangular array representations of the sequence $\{\mathbf{S}^{*}_{n}\}$. We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function $\tau ^{\mathbf{O}}_{1}$ which embodies the complexity of $\mathbf{O}$ itself. The trim, strong triangular array representation corresponding to the second of these is as close as possible to the representation of $\{\mathbf{S}_{n}\}$ provided by the $\mathbf{X}_{i}$.

#### Article information

Source
Illinois J. Math., Volume 63, Number 2 (2019), 219-257.

Dates
Revised: 3 April 2019
First available in Project Euclid: 1 August 2019

https://projecteuclid.org/euclid.ijm/1564646432

Digital Object Identifier
doi:10.1215/00192082-7768719

Mathematical Reviews number (MathSciNet)
MR3987496

Zentralblatt MATH identifier
07088306

#### Citation

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

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