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 ${\mathbf{S}}_n^{\ast }$ 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 ${\mathbf{X}}_i$.