BRANCHING PROCESSES WITH IMMIGRATION AND INTEGER-VALUED TIME SERIES

In this paper, we indicate how integer-valued autoregressive time series Ginar(d) of ordre d, d ≥ 1, are simple functionals of multitype branching processes with immigration. This allows the derivation of a simple criteria for the existence of a stationary distribution of the time series, thus proving and extending some results by Al-Osh and Alzaid (1), Du and Li (9) and Gauthier and Latour (11). One can then transfer results on estimation in subcritical multitype branching processes to stationary Ginar(d) and get consistency and asymptotic normality for the corresponding estimators. The technique covers autoregressive moving average time series as well.

[1]  J. Winnicki,et al.  Estimation of the variances in the branching process with immigration , 1991 .

[2]  Mohamed Alosh,et al.  FIRST‐ORDER INTEGER‐VALUED AUTOREGRESSIVE (INAR(1)) PROCESS , 1987 .

[3]  M. Quine,et al.  Estimation for multitype branching processes , 1977, Journal of Applied Probability.

[4]  Eugene Seneta,et al.  ESTIMATION THEORY FOR GROWTH AND IMMIGRATION RATES IN A MULTIPLICATIVE PROCESS , 1972 .

[5]  G. O'Brien.  Limit theorems for sums of chain-dependent processes , 1974, Journal of Applied Probability.

[6]  P. H. Leslie.  SOME FURTHER NOTES ON THE USE OF MATRICES IN POPULATION MATHEMATICS , 1948 .

[7]  Charles J. Mode,et al.  Multitype branching processes;: Theory and applications , 1971 .

[8]  Fw Fred Steutel,et al.  Discrete analogues of self-decomposability and stability , 1979 .

[9]  Peter A. W. Lewis,et al.  Discrete Time Series Generated by Mixtures. I: Correlational and Runs Properties , 1978 .

[10]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[11]  A. Alzaid,et al.  Integer-valued moving average (INMA) process , 1988 .

[12]  E. Seneta,et al.  Goodness-of-fit for a branching process with immigration using sample partial autocorrelations , 1989 .

[13]  P. H. Leslie.  On the use of matrices in certain population mathematics. , 1945, Biometrika.

[14]  E. G. Lewis.  On the Generation and Growth of a Population , 1977 .

[15]  M. Deistler,et al.  The linear model formulation of a multitype branching process applied to population dynamics. , 1974, Journal of the American Statistical Association.

[16]  E. McKenzie,et al.  Autoregressive moving-average processes with negative-binomial and geometric marginal distributions , 1986, Advances in Applied Probability.

[17]  E. Seneta,et al.  Independence of partial autocorrelations for a classical immigration branching process , 1991 .

[18]  J. H. Pollard,et al.  On the use of the direct matrix product in analysing certain stochastic population models , 1966 .

[19]  C. Z. Wei,et al.  Some asymptotic results for the branching process with immigration , 1989 .

[20]  K. N. Venkataraman.  A time series approach to the study of the simple subcritical Galton–Watson process with immigration , 1982, Advances in Applied Probability.

[21]  P. H. Leslie.  On the use of matrices in certain population mathematics. , 1945, Biometrika.

[22]  Mohamed Alosh,et al.  An integer-valued pth-order autoregressive structure (INAR(p)) process , 1990, Journal of Applied Probability.

[23]  Li Yuan,et al.  THE INTEGER‐VALUED AUTOREGRESSIVE (INAR(p)) MODEL , 1991 .

[24]  Ed McKenzie.  The distributional structure of finite moving-average processes , 1988 .

[25]  E. McKenzie,et al.  Some ARMA models for dependent sequences of poisson counts , 1988, Advances in Applied Probability.