Series convergence question

Question

This question occured to me in the context of ARMA time series analysis: Let $\alpha_n\geq 0,\, n= 1,2,\dots$ such that $\sum_{n=1}^\infty\alpha_n = 1$ and define the sequence $(\beta_n)_{n=0,1,\dots}$ recursively by $\beta_0:=1$ and $\beta_n:=\sum_{k=1}^n\alpha_k\beta_{n-k}$. Note that we have $$ \sum\limits_{n = 0}^\infty \beta_n z^n \Bigg(1-\sum\limits_{k=1}^\infty\alpha_kz^k\Bigg) = 1,\quad z\in\mathbb{C},|z|<1. $$ It is clear, that $\sum_{n=0}^\infty\beta_n =\infty$. But what about $\sum_{n=0}^\infty\beta_n^2$ ? Do these latter series diverge to infinity for any choice of $(\alpha_n)\subset \mathbb{R}_{\geq 0}$ with $\sum_{n=1}^\infty\alpha_n = 1$? Or are there specific (decay) conditions on $(\alpha_n)$ that would give $\sum_{n=0}^\infty\beta_n^2<\infty$? Any help is very much appreciated.