On Scale and Exponent Functions of a Class of Limiting Processes

Suppose {X_n, n ∈ Z₊) - see below for the notations used - is a sequence of random variables (rv's) such that, for every t > 0 and for ψ given by (1.1) below,

ψ(X_(nt);α_n;β_n)->Y(t),

where Y\l) (in particular) is (assumed) nondegenerate. Then it is shown that, for some positive functions α and β on (0,∞) and for every t > 0,

Y(t)=ψ(Y(1);α(t),β(t))

Explicit expressions for a and p are derived. As an application, the (Pancheva) class of all possible limit laws for power normalized partial maxima of i.i.d rv's ("the p-max stable laws") is re-derived.