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.