Let a linear regression be given. For detecting change-points, it is usual to consider the sequence of partial sums of least squares residuals whence a partial sums process is defined. Given a sequence of exact experimental designs, we consider for each design the corresponding partial sums process. If the sequence of designs converges to a continuous design, we derive the explicit form of the limit process of the corresponding sequence of partial sums processes. This is a complicated function of the Brownian motion. These results are useful for the problem of testing for change of regression at known or unknown times.
"A functional central limit theorem for regression models." Ann. Statist. 26 (4) 1398 - 1410, August 1998. https://doi.org/10.1214/aos/1024691248