On Derriennic's almost subadditive ergodic theorem
Let \(X=(X_{s,t})\subset L^ 1\) and \(h=(h_{s,t})\subset L^ 1_+\) be two stationary families of random variables satisfying the almost subadditivity condition \[ X_{s,u}-X_{s,t}-X_{t,u}\leq h_{t,u} \] for all integers \(0\leq s<t<u\). It is shown that \((n^{-1}X_{0,n})\) converges a.e. and in \(L^ 1\) as \(n\to \infty\) if the following three moment conditions are satisfied: \(\inf_{n\geq 1}n^{-1}E[X_{0,n}]>-\infty\); \(\lim_{n\to \infty}n^{-1}E[h_{0,n}]=0;\) \(\liminf_{n\to \infty}n^{-1}\sum^{n}_{i=1}E[h_{0,i}]<\infty.\) The a.e. convergence of \((n^{-1}X_{0,n})\) has been proved by \textit{Y. Derriennic} [Ann. Probab. 11, 669-677 (1983; Zbl 0586.28014)] under a slightly stronger moment condition on h. In the present paper it is also shown that \((n^{-1}X_{0,n})\) converges a.e. if X and h are merely supposed to be superstationary in the sense of \textit{T. Kamae}, \textit{U. Krengel} and \textit{G. L. O'Brien} [ibid. 5, 899-912 (1977; Zbl 0371.60013)] provided the above moment conditions are modified in an obvious way. This generalizes a result of \textit{M. Abid} [C. R. Acad. Sci., Paris, Sér. A 287, 149-152 (1978; Zbl 0386.60028)].
scientific article; zbMATH DE number 4066035