Inverse renewal thinning of Cox and renewal processes

A key result giving inter-connection amongst Bernstein function, completely monotone function and Stieltjes transform is established and used to derive sufficient conditions for the closure of the class of renewal processes under inverse renewal thinning. It is further proved that the class of Cox and renewal processes is closed under inverse renewal thinning for Bernoulli thinning, truncated negative binomial and truncated Poisson thinning.