English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, weanalyze the rate of convergence of $\mathbb{E}X_{0,n}/n$ to its limit $g$. Wedefine an exponent $\gamma$ given roughly by $\mathbb{E}X_{0,n} \sim ng +n^\gamma$, and, assuming existence of a fluctuation exponent $\chi$ that gives$\mathrm{Var}~X_{0,n} \sim n^{2\chi}$, we provide a lower bound for $\gamma$ ofthe form $\gamma \geq \chi$. The main requirement is that $\chi \neq 1/2$. Inthe case $\chi=1/2$ and under the assumption $\mathrm{Var}~X_{0,n} = O(n/(\logn)^\beta)$ for some $\beta>0$, we prove $\gamma \geq \chi - c(\beta)$ for a$\beta$-dependent constant $c(\beta)$. These results show in particular thatnon-diffusive fluctuations are associated to non-trivial $\gamma$. Variousmodels, including first-passage percolation, directed polymers, the minimum ofa branching random walk and bin packing, fall into our general framework, andthe results apply assuming $\chi$ exists. In the case of first-passagepercolation in $\mathbb Z^d$, we provide a version of $\gamma \geq -1/2$without assuming existence of $\chi$.

Rate of convergence of the mean for sub-additive ergodic sequences