On the age distribution of a Markov chain

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞). A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

[1]  B. Levikson.  The age distribution of Markov processes , 1977, Journal of Applied Probability.

[2]  T. Speed,et al.  Lagrange Distributions and their Limit Theorems , 1977 .

[3]  R. Phatarfod,et al.  Dams with additive inputs revisited , 1977, Journal of Applied Probability.

[4]  W. D. Kaigh.  A Conditional Local Limit Theorem for Recurrent Random Walk , 1975 .

[5]  A. Pakes,et al.  Conditional limit theorems for a left-continuous random walk , 1973, Journal of Applied Probability.

[6]  M. Serge Dubuc,et al.  La densité de la loi-limite d'un processus en cascade expansif , 1971 .

[7]  James H. Foster,et al.  A Limit Theorem for a Branching Process with State-Dependent Immigration , 1971 .

[8]  J. Williamson.  Random walks and Riesz kernels. , 1968 .

[9]  E. Seneta.  The Galton-Watson process with mean one , 1967, Journal of Applied Probability.

[10]  W. Pruitt.  Strong ratio limit property for $R$-recurrent Markov chains , 1965 .

[11]  S. Orey,et al.  Ratio limit theorems for Markov chains , 1964 .

[12]  A. Garsia.  Some Tauberian theorems and the asymptotic behavior of probabilities of recurrent events , 1963 .

[13]  S. Orey,et al.  Asymptotic behavior of successive coefficients of some power series , 1962 .

[14]  Adriano M. Garsia,et al.  A discrete renewal theorem with infinite mean , 1962 .

[15]  A. Pakes.  Some results for non-supercritical Galton-Watson processes with immigration , 1975 .

[16]  A. Pakes,et al.  A branching process with a state dependent immigration component , 1971, Advances in Applied Probability.

[17]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[18]  K. Bruce Erickson,et al.  Strong renewal theorems with infinite mean , 1970 .

[19]  S. Dubuc.  La fonction de Green d'un processus de Galton-Watson , 1970 .

[20]  E. Seneta.  Functional equations and the Galton-Watson process , 1969, Advances in Applied Probability.

[21]  J. Folkman,et al.  On Markov chains with the strong ratio limit property , 1965 .