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October, 1976 Multiparameter Subadditive Processes
R. T. Smythe
Ann. Probab. 4(5): 772-782 (October, 1976). DOI: 10.1214/aop/1176995983

Abstract

Let $N$ be the positive integers. We define a class of processes indexed by $N^r \times N^r$ which we call subadditive (when $r = 1$ our definition coincides with the usual one). Under a first moment condition we prove mean convergence of $x_{0t}/|\mathbf{t}|$ as each coordinate of $\mathbf{t} \rightarrow \infty$, where $|\mathbf{t}| = t_1 t_2 \cdots t_r$. If the process is strongly subadditive (a more restrictive condition) then the same first moment condition gives a.s. sectorial convergence. We conjecture (and verify in several cases) that an $L(\log L)^{r-1}$ integrability condition is sufficient to give unrestricted a.s. convergence.

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R. T. Smythe. "Multiparameter Subadditive Processes." Ann. Probab. 4 (5) 772 - 782, October, 1976. https://doi.org/10.1214/aop/1176995983

Information

Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0339.60022
MathSciNet: MR423495
Digital Object Identifier: 10.1214/aop/1176995983

Subjects:
Primary: 60F15
Secondary: 28A65 , 60G10

Keywords: ergodic theory , multiparameter processes , Subadditive processes

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 5 • October, 1976
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