Ruin probabilities by Padé’s method: simple moments based mixed exponential approximations (Renyi, De Vylder, Cramér–Lundberg), and high precision approximations with both light and heavy tails

We revisit below Padé and other rational approximations for ruin probabilities, of which the approximations mentioned in the title are just particular cases. We provide new simple Tijms-type and moments based approximations, and show that shifted Padé approximations are quite successful even in the case of heavy tailed claims.