Edgeworth correction for the largest eigenvalue in a spiked PCA model

We study improved approximations to the distribution of the largest eigenvalue $\widehat{\mathrm{\ell }}$ of the sample covariance matrix of $n$ zero-mean Gaussian observations in dimension $p+1$. We assume that one population principal component has variance $\mathrm{\ell }>1$ and the remaining `noise’ components have common variance $1$. In the high dimensional limit $p/n \to \gamma > 0$, we begin study of Edgeworth corrections to the limiting Gaussian distribution of $\widehat{\mathrm{\ell }}$ in the supercritical case $\ell > 1 + \sqrt \gamma$. The skewness correction involves a quadratic polynomial as in classical settings, but the coefficients reflect the high dimensional structure. The methods involve Edgeworth expansions for sums of independent non-identically distributed variates obtained by conditioning on the sample noise eigenvalues, and limiting bulk properties $\mathit{\text{and}}$ fluctuations of these noise eigenvalues.