Let \[ A(z)y_ k=C(z)w_ k,\quad y_ k=w_ k=0\quad\text{for}\;k\leq0, \] be a one-dimensional ARMA process, where \(A(z)=1+a_ 1z+\cdots+a_ pz^ p\) and \(C(z)=1+c_ 1z+\cdots+c_ rz^ r\) are coprime polynomials in backward shift operator \(z\) (\(zy_ k=y_ {k-1}\)), \(\{w_ k\}\) is a sequence of independent random variables with zero mean, \(\sup_ k E| w_ k| ^ \gamma0\). Assuming that \[ \limsup_ {n\to\infty}{1\over n}\sum_ {k=1}^ ny_ k^ 2<\infty, \] the authors prove that \(A(z)\) is stable, i.e., \(A(z)\neq0\) for \(| z| \leq1\).