Using the counting process equation, the paper analyses the estimation of transition probabilities p ij (s,t) for multistate life table, gives the estimations of the number of sample for each state and covariance, and discusses the asymptotic theory for this estimation.

Using the counting process techniques, this paper discussed the consistency and weak convergence of the estimation for expected length of sojourn time for multistate life table; and gave the corresponding covariance matrix estimator and its asymptotic properties.

Secondly, we give a new counting process, Compound PNB process.

In this paper, using martingale condition, we induce Lundberg Fundamental equation when the accumulated claim process is modled as a compound poisson process or a compound binomial process and the interest wave is a poisson process with jump.

The paper considers the negative risk model with the aggregate claims modeled as a compound binomial process.

Structural reliability design of equivalent exponential random variable for exponential distribution-stationary binomial process mode;

Considering the process of the stress is stationary binomial process and strength obeys the distribution of Erlang,gives the distribution of the maximum of stress and then obtains the UMVUE of structural reliability.

Considering the process of stress is stationary binomial process and strength obeys the distribution of exponential,gives the distribution of maximum of stress and then obtains the formulars of design and estimation of structural reliability.