This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chainwhich is irreducible and reversible outside the absorbing point.This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues.In the discrete time setting, if the underlying Dirichlet eigenvalues(namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point)are nonnegative, we show that T is distributed as a mixture ofsums of independent geometric laws whose parameters are successiveDirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law.This result leads to a probabilistic interpretation of the spectrum,in terms of strong random times and local equilibria through a simple intertwining relation.Next this study is extended to the continuous time framework,wheregeometric laws have to be replaced by exponential distributions having the (opposite)Dirichlet eigenvalues of the generator as parameters.Returning to the discrete time setting we consider the influence ofnegative eigenvalues which are given another probabilistic meaning.These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

Linear combinations of exponential distribution functions are considered, and the class of distribution functions so obtainable is investigated. Convex combinations correspond to hyperexponential distributions, while non-convex combinations yield, among other, generalized Erlang distributions obtainable as sums of independent exponential random variables with different parameters.

For a given number n of different exponential distributions, the class investigated is an (n – 1)-dimensional convex subset of the n-dimensional real vector space generated by the n distribution functions. The geometric aspect of this subset is revealed, together with the location of hyperexponential and generalized Erlang distributions.

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.