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Polynomial time relatively computable triangular arrays for almost sure convergence. (English) Zbl 07088306
Summary: 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 \(\mathbf{S}_n:=\sum_{i=1}^n \mathbf{X}_i\). We let \(\mathbf{S}_n^\ast\) 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 \(\mathbf{S}_n^\ast\) 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 \(\{0,\ldots,2^{n(M+1)}-1\}\), 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 \(\{S_n^\ast\}\). 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 \(\{\mathbf{S}_n\}\) provided by the \(\mathbf{X}_i\).
60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems
60F05 Central limit and other weak theorems
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q25 Analysis of algorithms and problem complexity
Full Text: DOI arXiv Euclid
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