Asymptotic behavior of unstable INAR(p) processes

Abstract

In this paper the asymptotic behavior of an unstable integer-valued autoregressive model of order p (INAR( p)) is described. Under a natural assumption it is proved that the sequence of appropriately scaled random step functions formed from an unstable INAR( p) process converges weakly towards a squared Bessel process. We note that this limit behavior is quite different from that of familiar unstable autoregressive processes of order p. An application for Boston armed robberies data set is presented.

Statistical inference of 2-type critical Galton–Watson processes with immigration

“…In fact, in Ispány and Pap [12, Theorem 3.1], the above result has been prooved under the higher moment assumptions E( is nonnegative for all t ∈ R + with probability one, hence Z + t may be replaced by Z t under the square root in (3.8), see, e.g., Barczy et al [4,Remark 3.3].…”

“…By the method of the proof of X (n) D −→ X in Theorem 3.1 in Barczy et al [4], applying Lemma C.2, one can easily derive…”

“…By page 603 in Barczy et al [4], the mappings ψ n , n ∈ N, and ψ are measurable (the latter one is continuous too), since the coordinate functions are measurable. Using page 604 in Barczy et al [4], we obtain that the set…”

Test Statistics for Parameter Changes in INAR(p) Models and a Simulation Study

“…This is is the nearly unstable case, and in the unstable case the process loses its stationarity, E(X k ) → ∞ linearly and Theorem 2.2 is no longer valid-see e.g. Barczy, Ispány, and Pap (2011) This case is illustrated by Figure 3, where we can see that a small change in the coefficient can be accompanied by a large change in the innovation without any visible change occurring. In Figure 3 the coefficient and the innovation mean are constant throughout the trajectory.…”

Statistical inference for critical continuous state and continuous time branching processes with immigration

“…By continuous mapping theorem (see, e.g., the method of the proof of X (n) D −→ X in Theorem 3.1 in Barczy et al 2011), one can easily derive…”