A Proof of the Subadditive Ergodic Theorem

2017 ◽  
pp. 343-354
Author(s):  
Anders Karlsson
Keyword(s):  
Ergodic Theorem ◽  
Subadditive Ergodic Theorem

scholarly journals An Improved Subadditive Ergodic Theorem

1985 ◽  
Vol 13 (4) ◽  
pp. 1279-1285 ◽  
Author(s):  
Thomas M. Liggett
Keyword(s):  
Ergodic Theorem ◽  
Subadditive Ergodic Theorem

scholarly journals THE RANGE OF TREE-INDEXED RANDOM WALK

2014 ◽  
Vol 15 (2) ◽  
pp. 271-317 ◽  
Author(s):  
Jean-François Le Gall ◽  
Shen Lin
Keyword(s):  
Random Walk ◽  
Ergodic Theorem ◽  
Random Tree ◽  
The Other ◽  
Branching Random Walk ◽  
Initial Size ◽  
Symmetric Random Walk ◽  
Dimensional Lattice ◽  
Exponential Moments ◽  
Subadditive Ergodic Theorem

We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.

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