Exponential tilt of the ruin renewal kernel ($k_R(x)=(\lambda /c)e^{Rx}\overline{F}(x)$)

The ultimate ruin probability satisfies a tail-forced defective renewal equation. Multiplying it by $e^{Ru}$ at an adjustment coefficient turns its kernel into the probability density $k_R$. Normalization is exactly $\lambda (M(R)-1)=cR$. The density is absolutely continuous even when the original claim law has atoms; thus it has a nonarithmetic distribution.