A family of matrices, the discretized brownian bridge, and distance-based regression

The investigation of a distance-based regression model, using a one-dimensional set of equally spaced points as regressor values and √x − y as a distance function, leads to the study of a family of matrices which is closely related to a discrete analog of the Brownian-bridge stochastic process. We describe its eigenstructure and several properties, recovering in particular well-known results on tridiagonal Toeplitz matrices and related topics.