  We consider random coefficient autoregressive models of infinite order
(AR($\infty$)) under the assumption of non-negativity of the coefficients. We
develop novel methods yielding sufficient or necessary conditions for
finiteness of moments, based on combinatorial expressions of first and second
moments. The methods based on first moments recover previous sufficient
conditions by Doukhan and Wintenberger in our setting. The second moment method
provides in particular a necessary and sufficient condition for finiteness of
second moments which is different, but shown to be equivalent to the classical
criterion of Nicholls and Quinn in the case of finite order. We further
illustrate our results through two examples.
