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Rate of convergence of the mean of sub-additive ergodic processes

Friday 20th March 2015 - 11:30 to 12:30
INI Seminar Room 1
Co-authors: Michael Damron (Indiana University), Jack T. Hanson (Indiana University)

For a subadditive ergodic sequence ${X_{m,n}}$, Kingman's theorem gives convergence for the terms $X_{0,n}/n$ to some non-random number $g$. In this talk, I will discuss the convergence rate of the mean $\mathbb EX_{0,n}/n$ to $g$. This rate turns out to be related to the size of the random fluctuations of $X_{0,n}$; that is, the variance of $X_{0,n}$, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent. The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Michael Damron and Jack Hanson.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons