An improved subadditive ergodic theorem
Let \(\{X_{m,n}\}\) \((m=0,1,2,...,n-1\), \(n=1,2,...)\) be a double array of random variables satisfying (i) \(X_{0,n}\leq X_{0,m}+X_{m,n}\) \((0<m<n)\), (ii) the joint distribution of \(\{X_{m+1,m+k+1};k\geq 1\}\) is the same as those of \(\{X_{m,m+k};k\geq 1\}\) for each \(m\geq 0\), (iii) for each \(k\geq 1\) \(\{X_{nk,(n+1)k};n\geq 1\}\) is a stationary process. The author proves (i) \(\gamma =\lim n^{-1}EX_{0,n}=\inf n^{- 1}EX_{0,n}\), (ii) \(X=\lim n^{-1}X_{0,n}\) exists a.s. and in \(L_ 1\), (iii) \(EX=\gamma\), (iv) if the stationary processes in condition (iii) are ergodic then \(X=\gamma\) a.s. This result is essentially the same as the corresponding result of \textit{J. F. C. Kingman} [ibid. 1, 883-909 (1973; Zbl 0311.60018)] however the conditions are much weaker than the corresponding conditions of Kingman.
scientific article; zbMATH DE number 3927992