Theorem (KMT, 1975) On a suitable
probability space for independent uniform (0,1) r.v. the empirical process can be approximated by a sequence of Brownian bridges such that
for all positive integers n and all , where a, b, and c are positive constants.
Corollary
A corollary of that theorem is that for any real
iid r.v. with
cdf it is possible to construct a probability space where independent[clarification needed] sequences of empirical processes and
Gaussian processes exist such that
Komlos, J., Major, P. and Tusnady, G. (1975) An approximation of partial sums of independent rv鈥檚 and the sample df. I, Wahrsch verw Gebiete/Probability Theory and Related Fields, 32, 111鈥131.
doi:
10.1007/BF00533093
Komlos, J., Major, P. and Tusnady, G. (1976) An approximation of partial sums of independent rv鈥檚 and the sample df. II, Wahrsch verw Gebiete/Probability Theory and Related Fields, 34, 33鈥58.
doi:
10.1007/BF00532688