Defective renewal equation
= Defective renewal equation
{title2=$z=g+z*Q,\quad Q([0,\infty))<1$}
A <Defective renewal equation> is $z(u)=g(u)+\int_{[0,u]}z(u-x)Q(dx)$ where the nonnegative kernel measure has total mass less than one. Its <renewal representation> is a convergent sum of <convolution>[convolutions]. <Exponential tilting> can turn the kernel into a <probability measure>, enabling the <key renewal theorem>.