Let
X 1,
X 2,… be a sequence of random variables. Let
S k =
X 1+
???+
X k and assume that
S k /
b k converges in distribution for some numerical sequence (
b k ). We study the weak convergence of the random processes {Λ
n (
z),
z∈?}, where
$Lambda_{n}(z)=frac{1}{n}sum_{k=1}^{n}Ileft{frac{S_{k}}{b_{k}}leq zright}.$
We consider the same problem when the normalized partial sums
S k /
b k are replaced by other functionals of the sequence (
X n ). In particular, we investigate the case of sample extremes in detail.