Generalised regular variation of arbitrary order

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h 6≡ 0 and g > 0 such that f(xt)− f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) leads eventually to a relation of the form f(xt)− f(t) = h1(x)g1(t) + · · ·+ hn(x)gn(t) + o(gn(t)), each gi being of smaller order than its predecessor gi−1. The function f is said to be generalised regularly varying of order n with rate vector g = (g1, . . . , gn) ′. Under general assumptions, g itself must be regularly varying in the sense that g(xt) = xg(t) + o(gn(t)) for some upper triangular matrix B ∈ R, and the vector of limit functions h = (h1, . . . , hn) is of the form h(x) = c ∫ x 1 u udu for some row vector c ∈ R. The usual results in the theory of regular variation such as uniform convergence and Potter bounds continue to hold. An interesting special case arises when all the rate functions gi are slowly varying, yielding Π-variation of order n, the canonical case being that B is equivalent to a single Jordan block with zero diagonal. The theory is applied to a long list of special functions.