On the asymptotic behavior of the ruin probability for an infinite period when the epochs of claims form a renewal process

Summary In two previous papers 11 and 12 the author i.a. proved generalizations of Cramer's classical 4 asymptotic formula for the ruin probability for an infinite period proved by him when the epochs of claims form a Poisson process. However, these generalizations relied on a restriction as to the distribution function, K(t), t ⩾ 0, K(O) = 0, for the interoccurrence times between successive claims. In the present paper this restriction is relaxed for non-arithmetic F(x) = ∫∞0 P(x + ct)dK(t). For arithmetic F(x) a slightly modified asymptotic formula is proved. Here P(y), − ∞ < y < ∞ denotes the distribution function of the claim amounts and c denotes the gross risk premium per time unit. Of course, the restrictions on c and P(y) and −for c<0—on K(t) necessary for the existence of the positive constant R are still assumed. However, for functions not satisfying these conditions it is sometimes possible to give other asymptotic formulas. An example is given enclosing the case when P(y) is of Pareto type.