We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point β. When α is badly approximable and β is badly approximable with respect to α, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when α is quadratic irrational, β is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of β, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

Abstract. We prove bounded rational ergodicity for some discrepancy skew products whose rotation number has bad rational approximation. This is done by considering the asymptotics of associated affine random walks.

Let {Xn} be an integer valued Markov Chain with finite state space. Let Sn = n k=0 X k and let Ln (x) be the number of times S k hits x ∈ Z up to step n. Define the normalized local time process ln (t, x) byThe subject of this paper is to prove a functional, weak invariance principle for the normalized sequence ln (t, x) ,i.e.we prove under the assumption of strong aperiodicity of the Markov Chain that the normalized local times converge in distribution to the local time of the Brownian Motion.

Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones. They are trained on benchmarks with a fixed number of data points per task. This number is usually arbitrary and it is unknown how it affects performance at testing. Since labelling of data is expensive, finding the optimal allocation of labels across training tasks may reduce costs. Given a fixed budget of labels, should we use a small number of highly labelled tasks, or many tasks with few labels each? Should we allocate more labels to some tasks and less to others? We show that: 1) If tasks are homogeneous, there is a uniform optimal allocation, whereby all tasks get the same amount of data; 2) At fixed budget, there is a trade-off between number of tasks and number of data points per task, with a unique solution for the optimum; 3) When trained separately, harder task should get more data, at the cost of a smaller number of tasks; 4) When training on a mixture of easy and hard tasks, more data should be allocated to easy tasks. Interestingly, Neuroscience experiments have shown that human visual skills also transfer better from easy tasks. We prove these results mathematically on mixed linear regression, and we show empirically that the same results hold for few-shot image classification on CIFAR-FS and mini-ImageNet. Our results provide guidance for allocating labels across tasks when collecting data for meta-learning.

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk. §0 Introduction A rational step function is a right continuous, step function on the additive circle T ∶= R Z ≅ [0, 1) taking values in R d , whose discontinuity points are rational.Let ϕ ∶ T → R d be a rational step function.The skew productsT α,ϕ (x, y) ∶= (x + α, y + ϕ(x)) are conservative if and only if T ϕ(t)dt = 0.Necessity follows from the ergodic theorem and sufficiency follows from the Denjoy-Koksma inequality (see below).Consider the collections of badly approximable irrationals