This paper obtains the asymptotic representation of the supremum of a class of functionals of the empirical distribution, with application to estimating the strength of a bundle of parallel filaments, when the observations are strongly dependent. Unlike the case of weakly dependent observations discussed by P. K. Sen and B. B. Bhattacharyya [Z. Wahrsch. Verw. Gebiete 34 (1976), no. 2, 113–118], the limiting distributions of these functionals are not always normal. The nature of the limiting distribution depends heavily on the Hermite rank of a class of indicator functions, and the rate of convergence is much slower in this case (compared to the case of weak dependence). A law of iterated logarithm (LIL) for these functionals is also derived.