The sample paths of cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, when appropriately normalized, converge weakly (as $n$ increases indefinitely) to the sum of a Brownian motion with time change and an integrated Brownian bridge which is independent of the Brownian motion. Applications in regression analysis are given.
"An Invariance Principle in Regression Analysis." Ann. Statist. 4 (3) 621 - 624, May, 1976. https://doi.org/10.1214/aos/1176343468