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Asymptotic behavior of unstable INAR(p) processes. (English) Zbl 1241.62122

The authors investigate the asymptotic properties of integer-valued autoregressive model INAR(p) determined by the equation

X k =α 1 X k-1 ++α p X k-p +ε k ,k,

where αX= j=1 X ξ j if X>0, αX=0 if X>0, ξ j ,j, are i.i.d. Bernoulli random variables with mean α[0,1], ε k are i.i.d. non-negative integer-valued random variables with the mean μ ε , and α 1 ,,α p [0,1]. Under the assumption that the second moment of the innovation ε k is finite the authors proved that the sequence of appropriately scaled random step functions X t n =X [nt] /n, t + , n, formed from an unstable INAR(p) process (α 1 ++α p =1) converges weakly towards a squared Bessel process determined by the stochastic differential equation dX t =μ ε dt+σ α 2 X t + dW t /ϕ ' (1), X 0 =0, t + , where ϕ ' (1)=α 1 +2α 2 ++pα p >0, σ α 2 =α 1 (1-α 1 )+2α 2 ++pα p (1-α p ), W t ,t + , is a standard Wiener process. This limit process is a continuous branching process also known as square-root process or Cox-Ingersoll-Ross process. This asymptotic behavior of unstable INAR(p) models is quite different from that of familiar (real-valued) unstable autoregressive processes of order p (AR(p) models). An application to Boston armed robberies data is presented.

MSC:
62M10Time series, auto-correlation, regression, etc. (statistics)
60J80Branching processes
60F99Limit theorems (probability)
62P25Applications of statistics to social sciences
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