Uncorrelated, but Not Independent: What volatility does to estimated correlations

Observations can be uncorrelated while their estimated correlations have dependent errors. Beginning with two fair coin games, this paper explains how sign-dependent volatility affects the joint uncertainty of estimated autocovariances. An explicit construction gives two stationary processes with the same one-observation distribution, zero autocorrelation at every positive lag, and the same complete independent squared-process law, yet different covariance matrices for their autocovariance estimates. We derive the construction’s covariance matrix, establish sharp two-model risk bounds for procedures using only squared observations, and demonstrate exact recovery from signed observations within the specified model family. These results are presented within the established framework of weighted-Gram covariance estimation. Eleven figures, worked examples, and exercises introduce the ideas progressively. The integrated technical supplement provides proofs, finite-record boundary corrections, fixed-dimensional inference results, and reproducible simulations. This deposit contains the manuscript, LaTeX sources, figures, computational code, numerical results, and reproduction instructions.