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An improved subadditive ergodic theorem. (English) Zbl 0579.60023
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 J. F. C. Kingman [ibid. 1, 883-909 (1973; Zbl 0311.60018)] however the conditions are much weaker than the corresponding conditions of Kingman.
Reviewer: P.Révész

##### MSC:
 60F15 Strong limit theorems 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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