On stochastic structure of thinning-based
non-negative vector-valued ARMA processes

M. Ispány

Faculty of Informatics, University of Debrecen, Hungary

Count time series models are widely used in practice, see the survey [1]. In the talk, we provide a unified, distribution-free discussion of the following thinning-based non-negative integer vector-valued ARMA model

$$Y_k=\sum _{i=0}^p{A}_i^k\circ Y_{k-i}+\sum _{j=0}^q{B}_j^k\circ {\epsilon }_{k-j},k\in \mathbb{Z},$$ (1)

where $\{A_i^k\circ \}$, $i=0,1,\mathrm{\dots },p$, and $\{B_j^k\circ \}$, $j=0,1,\mathrm{\dots },q$, are independent identically distributed matricial thinning operators of dimension $d\times d$, respectively, and the input process $\{{\epsilon }_k\}$ is a sequence of independent identically distributed ${\mathbb{N}}_0^d$-valued random vectors. The matricial thinning operators and the random vectors are mutually independent. The solution $\{Y_k\}$ of (1) is a generalized branching process called INVARMA$(p,q)$ process, see [2] for an example.

We give a simple moment assumption and a spectral criterion involving the model parameters that ensures the existence and uniqueness of a solution to (1). We present a graphical model for describing the stochastic structure of the process and the dependencies among offspring generations and the moving average part of the model as a one-sided infinite skew comb.

Joint work with P. Bondon (CentraleSupélec, France), V. A. Reisen (UFES, Brazil).

Keywords: ARMA model, count time series, graphical model.