A Central Limit Theorem for Vincular Permutation Patterns

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.

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References in corpus (3)

• Stein couplings for normal approximation
• Pattern Avoidance for Random Permutations
• The probability of avoiding consecutive patterns in the Mallows distribution

Cited by in corpus (8)

• Counting Small Permutation Patterns
• Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels
• Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces
• Central limit theorems for patterns in multiset permutations and set partitions
• The volume of simplices in high-dimensional Poisson-Delaunay tessellations
• A geometric approach to conjugation-invariant random permutations
• Moments of permutation statistics and central limit theorems
• Asymptotic normality for $m$-dependent and constrained $U$-statistics, with applications to pattern matching in random strings and permutations