  We derive theorems which outline explicit mechanisms by which anomalous
scaling for the probability density function of the sum of many correlated
random variables asymptotically prevails. The results characterize general
anomalous scaling forms, justify their universal character, and specify
universality domains in the spaces of joint probability density functions of
the summand variables. These density functions are assumed to be invariant
under arbitrary permutations of their arguments. Examples from the theory of
critical phenomena are discussed. The novel notion of stability implied by the
limit theorems also allows us to define sequences of random variables whose sum
satisfies anomalous scaling for any finite number of summands. If regarded as
developing in time, the stochastic processes described by these variables are
non-Markovian generalizations of Gaussian processes with uncorrelated
increments, and provide, e.g., explicit realizations of a recently proposed
model of index evolution in finance.
