$L^p$ bounds for a central limit theorem with involutions

Abstract

Let $E=((e_{ij}))_{n\times n}$ be a fixed array of real numbers such that $e_{ij}=e_{ji}, e_{ii}=0$ for $1\le i,j \le n$. Let the permutation group be denoted by $S_n$ and the collection of involutions with no fixed points by $Π_n$, that is, $Π_n=\{π\in S_n: π^2= id, π(i)\neq i\,\forall i\}$ with id denoting the identity permutation. For $π$ uniformly chosen from $Π_n$, let $Y_E=\sum_{i=1}^n e_{iπ(i)}$ and $W=(Y_E-μ_E)/σ_E$ where $μ_E=E(Y_E)$ and $σ_E^2= Var(Y_E)$. Denoting by $F_W$ and $Φ$ the distribution functions of $W$ and a $\mathcal{N}(0,1)$ variate respectively, we bound $||F_W-Φ||_p$ for $ 1\le p\le \infty$ using Stein's method and the zero bias transformation. Optimal Berry-Esseen or $L^\infty$ bounds for the classical problem where $π$ is chosen uniformly from $S_n$ were obtained by Bolthausen using Stein's method. Although in our case $π\in Π_n$ uniformly, the $L^p$ bounds we obtain are of similar form as Bolthausen's bound which holds for $p=\infty$. The difficulty in extending Bolthausen's method from $S_n$ to $Π_n$ arising due to the involution restriction is tackled by the use of zero bias transformations.