  The scale invariant properties of wave functions in finite samples of one
dimensional random systems with correlated disorder are analyzed. The random
dimer model and its generalizations are considered and the wave functions are
compared. Generalized entropic localization lengths are introduced in order to
characterize the states and compared with their behavior for exponential
localization. An acceptable agreement is obtained, however, the exponential
form seems to be an oversimplification in the presence of correlated disorder.
According to our analysis in the case of the random dimer model and the two new
models the presence of power-law localization cannot be ruled out.
