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Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with tails of the form $ O(n^{-\beta}) $, $ \beta>2 $. We prove optimal moment bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $ \sum_{0\le i5$. Our method of proof is as follows; (i) prove that $ T $ satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimates. Such iterated moment bounds arise when using rough path theory to prove deterministic homogenisation results. Indeed, by a recent result of Chevyrev, Friz, Korepanov, Melbourne & Zhang we have convergence an It\^o diffusion for fast-slow systems of the form \[ x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad y_{k+1}=T y_k \] in the optimal range $ \beta>2. $, Comment: 25 pages. Minor changes. To appear in Communications in Mathematical Physics

Research from the wars in Afghanistan and Iraq have focused on traumatic brain injury (TBI) and mental health conditions; however, it is becoming clear that other health concerns, such as respiratory illnesses, warrant further scientific inquiry. Early reports from theater and postdeployment health assessments suggested an association with deployment-related exposures (e.g., sand, burn pits, chemical, etc.) and new-onset respiratory symptoms. We used data from Veterans Affairs medical encounters between fiscal years 2003 and 2011 to identify trends in chronic obstructive pulmonary disease, asthma, and interstitial lung disease in veterans. We used data from Veterans Affairs and Department of Defense sources to identify sociodemographic (age, sex, race), military (e.g., service branch, multiple deployments) and clinical characteristics (TBI, smoking) of individuals with and without chronic lung diseases. Generalized estimating equations found significant increases over time for chronic obstructive pulmonary disease and asthma in both unadjusted and adjusted analyses. Trends for interstitial lung disease were significant only in adjusted analyses. Age, smoking, and TBI were also significantly associated with chronic lung diseases; however, multiple deployments were not associated. Research is needed to identify which characteristics of deployment-related exposures are linked with chronic lung disease.