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Suppose $(f,mathcal{X},mu)$ is a measure preserving dynamical system and $phi colon mathcal{X} to mathbb{R}$ a measurable function. Consider the maximum process $M_n:=max{X_1 ldots,X_n}$, where $X_i=phicirc f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $mu(X_1>u_n)to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $mu(M_nleq u_n,textrm{i.o.})$ depending on the sequence $u_n$. Specific examples are piecewise expanding interval maps including the Gauss map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.
We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain abstract results on almost sure exponential, stretched exponential and polynomial correlation decay
We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (linear case) and $widetilde F_{n+2} = |lambda widetilde F_{n+1} pm widetilde F_{n}|$ (non-linear case),
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether
Across the world, scholars are racing to predict the spread of the novel coronavirus, COVID-19. Such predictions are often pursued by numerically simulating epidemics with a large number of plausible combinations of relevant parameters. It is essenti
We investigate the convergence and convergence rate of stochastic training algorithms for Neural Networks (NNs) that, over the years, have spawned from Dropout (Hinton et al., 2012). Modeling that neurons in the brain may not fire, dropout algorithms