Estimate of Time Needed for a Coordinate of a Bernoulli Scheme to Fall into the First Column of a Young Tableau

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(e) = C/μ5) such that, given any n ∈ ℕ and e ∈ (0, 1), one can choose a measurable set $${{\mathcal{X}}_{n,\varepsilon }} \subset {\mathcal{X}}$$ of measure at least 1 − e 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(e)n2 insertions of the RSK (Robinson-Schensted-Knuth) algorithm.