Almost subadditive multiparameter ergodic theorems (Q1115002)

Multiparameter subadditive processes are processes indexed by suitable subsets A of \({\mathbb{Z}}^ d\) or \({\mathbb{R}}^ d\) and satisfying \(X_{A\cup B}\leq X_ A+X_ B\) for disjoint A, B. \textit{J. Moulin-Ollagnier} [Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. B 19, 257-266 (1983; Zbl 0524.28024)] and \textit{Y. Derriennic} [Ann. Probab. 11, 669-677 (1983; Zbl 0586.28014)] studied (for \(d=1)\) almost subadditive processes, i.e. processes which satisfy such an inequality only up to error terms which are comparatively small for large A. The present paper extends these studies to the case of general \(d\in {\mathbb{N}}.\) First, an \(L_ 1\)-convergence result is proved for stationary processes with a rather weak almost subadditivity property. This is applied to the travelling salesman problem for stationary integrable point processes. The main purpose of the paper is to extend results of Derriennic, \textit{T. M. Liggett} [ibid. 13, 1279-1285 (1985; Zbl 0579.60023)] and others on a.e.-convergence of \(X_ A/card(A)\) for processes satisfying a strong almost subadditivity condition and having certain monotonicity properties. The proof is based on subtle multiparameter analogues of a construction of \textit{R. Durrett} [see ibid. 12, 999-1040 (1984; Zbl 0567.60095), and ibid. 8, 890-907 (1980; Zbl 0457.60082)]. The precise statement of the results is too long to be included here.