Almost subadditive extensions of Kingman's ergodic theorem (Q1180571)

Let \(X=(X_ I)\) and \(U=(U_ I)\) be jointly stationary processes indexed by subintervals \(I\) of \(\mathbb{Z}^ +\), and \(U\geq 0\), Put \(\bar X_{[i,j[}=X_{[i,j[}/(j-i)\). \(X\) is called \((AS)\)-subadditive with respect to \(U\) if for any disjoint intervals \(J_ 1,J_ 2,\dots,J_ k\) such that their union \(J\) is an interval \(X_ J\leq\sum (X_{J_ i} + U_{J_ i})\). Assume that, in addition, \((m_ j)\) is an increasing sequence with \(\lim \bar U_{[0,m_ j[} = 0\) a.s., that \(X_{[0,1[}^ +\) is integrable and \(\lim\inf\bar X_{[0,j+m_ n[} \geq \lim\inf\bar X_{[0,m_ n[}\) a.s. for any \(j\geq1\). Theorem 1 asserts that under these conditions \(\lim\bar X_{[0,m_ n[}\) exists a.e.. A second theorem shows almost sure convergence of \(\bar X_{[0,n[}\) under a slightly stronger subadditivity condition under weak moment conditions on \(U\).