## Illinois Journal of Mathematics

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

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

Vladimir Dobrić†, Patricia Garmirian, Marina Skyers, and Lee J. Stanley

#### 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 ${\mathbf{S}}_{n}^{\ast}$ be the *quantile* of ${\mathbf{S}}_{n}$. If $\mathbf{O}$ is Rademacher, the sequence $\left\{{\mathbf{S}}_{n}\right\}$ 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 ${\mathbf{X}}_{1},\dots ,{\mathbf{X}}_{n}$ provide a *representation* of ${\mathbf{S}}_{n}$ that is *strong* in that their sum is equal to ${\mathbf{S}}_{n}$ pointwise. They represent ${\mathbf{S}}_{n}^{\ast}$ only in distribution. Are there strong representations of ${\mathbf{S}}_{n}^{\ast}$? 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 $\{0,\dots ,{2}^{n(M+1)}-1\}$, with the property that we call *admissibility*. Passing to sequences, $\left\{{\pi}_{n}\right\}$, of admissible permutations, these provide a complete classification of *trim, strong triangular array representations of the sequence* $\left\{{\mathbf{S}}_{n}^{\ast}\right\}$. We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function ${\tau}_{1}^{\mathbf{O}}$ 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 $\left\{{\mathbf{S}}_{n}\right\}$ provided by the ${\mathbf{X}}_{i}$.

#### Article information

**Source**

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

**Dates**

Received: 29 December 2016

Revised: 3 April 2019

First available in Project Euclid: 1 August 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00192082-7768719

**Mathematical Reviews number (MathSciNet)**

MR3987496

**Zentralblatt MATH identifier**

07088306

**Subjects**

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]

#### 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