A Central Limit Theorem for Vincular Permutation Patterns

TL;DR

This paper proves a central limit theorem for the number of occurrences of vincular permutation patterns in random permutations, establishing asymptotic normality and convergence rates using dependency graphs and variance estimation techniques.

Contribution

It introduces a novel recursive variance estimation method and applies dependency graph techniques to vincular permutation patterns, advancing understanding of their probabilistic behavior.

Findings

Number of vincular pattern occurrences is asymptotically normal.

Established convergence rates for the normal approximation.

Developed a recursive variance lower bound technique.

Abstract

We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem, we use the method of dependency graphs. The main difficulty is then to estimate the variance of our statistics. We need a lower bound on the variance, for which we introduce a recursive technique based on the law of total variance.