Invariant density for a class of initial distributions under quadratic mapping

For the discrete-time quadratic map xt+1=4xt(1-xt) the evolution equation for a class of non-uniform initial densities is obtained. It is shown that in the t to infinity limit all of them approach the invariant density for the map.

Properties of Equilibrium Distributions of Order n

The present work is intended to discuss various properties and reliability aspects of higher order equilibrium distributions in continuous, discrete and multivariate cases, which contribute to the study on equilibrium distributions. At first, we have to study and consolidate the existing literature on equilibrium distributions. For this we need some basic concepts in reliability. These are being discussed in the 2nd chapter, In Chapter 3, some identities connecting the failure rate functions and moments of residual life of the univariate, non-negative continuous equilibrium distributions of higher order and that of the baseline distribution are derived. These identities are then used to characterize the generalized Pareto model, mixture of exponentials and gamma distribution. An approach using the characteristic functions is also discussed with illustrations. Moreover, characterizations of ageing classes using stochastic orders has been discussed. Part of the results of this chapter has been reported in Nair and Preeth (2009). Various properties of equilibrium distributions of non-negative discrete univariate random variables are discussed in Chapter 4. Then some characterizations of the geo- metric, Waring and negative hyper-geometric distributions are presented. Moreover, the ageing properties of the original distribution and nth order equilibrium distribu- tions are compared. Part of the results of this chapter have been reported in Nair, Sankaran and Preeth (2012). Chapter 5 is a continuation of Chapter 4. Here, several conditions, in terms of stochastic orders connecting the baseline and its equilibrium distributions are derived. These conditions can be used to rede_ne certain ageing notions. Then equilibrium distributions of two random variables are compared in terms of various stochastic orders that have implications in reliability applications. In Chapter 6, we make two approaches to de_ne multivariate equilibrium distribu- tions of order n. Then various properties including characterizations of higher order equilibrium distributions are presented. Part of the results of this chapter have been reported in Nair and Preeth (2008). The Thesis is concluded in Chapter 7. A discussion on further studies on equilib- rium distributions is also made in this chapter.