Regular variation of the tail of a subordinated probability distribution

Abstract

Let F be a probability measure on the real line and G = Σ C ( k ) F k ∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V ( x ) vary regularly of order p for x → ∞ and either (1) V ( x ) F [ x , ∞)→ α ≧ 0 or (2) V ( x ) C [ x , ∞)→ γ ≧ 0. If ρ > 1 and F (–∞, 0) = 0, necessary and sufficient in order that V ( x ) G [ x , ∞)→ b , is that both (1) and (2) hold for suitable α and γ . For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.