Abstract
The paper considers the classical Bernoulli scheme, that is, a sequence of independent random variables identically distributed with respect to the Lebesgue measure m on the interval [0,1]. The space of realizations of this scheme is the infinite-dimensional cube \({\mathcal{X}} = ({[0,1]^{\mathbb{N}}},\mu )\) with Lebesgue measure μ = mℕ. It is proved that there exists a function k(·): (0, 1) → ℝ (which can be defined by k(ε) = C/μ5) such that, given any n ∈ ℕ and ε ∈ (0, 1), one can choose a measurable set \({{\mathcal{X}}_{n,\varepsilon }} \subset {\mathcal{X}}\) of measure at least 1 − ε so that the coordinate xn of any realization \(x = {{\rm{\{ }}{x_n}{\rm{\} }}_n} \in {{\mathcal{X}}_{n,\varepsilon }}\) reaches the first column of the Young P-tableau after at most k(ε)n2 insertions of the RSK (Robinson-Schensted-Knuth) algorithm.
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Funding
The authors thank the V. A. Rokhlin Scholarship Foundation and Chebyshev Laboratory, Department of Mathematics and Computer Science, St. Petersburg State University.
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Russian Text © The Author(s), 2020, published in Funktsional’nyi Analiz i Ego Prilozheniya, 2020, Vol. 54, No. 2, pp. 78–84.
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Azangulov, I.F., Ovechkin, G.V. Estimate of Time Needed for a Coordinate of a Bernoulli Scheme to Fall into the First Column of a Young Tableau. Funct Anal Its Appl 54, 135–140 (2020). https://doi.org/10.1134/S0016266320020069
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DOI: https://doi.org/10.1134/S0016266320020069