In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R d and undergoing a binary, supercritical branching with a constant rate λ > 0. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalisation fall into three qualitatively different classes. In, what we call, the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalisation is the square root of the size of the system. In the critical case the limit is still Gaussian, however the normalisation requires an additional term. Finally, when branching has large rate the situation is completely different. The limit is no longer Gaussian, the normalisation is substantially larger than the classical one and the convergence holds in probability.We prove also that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process). MSC: primary 60F05; 60J80 secondary 60G20

In the NIPS 2017 Learning to Run challenge, participants were tasked with building a controller for a musculoskeletal model to make it run as fast as possible through an obstacle course. Top participants were invited to describe their algorithms. In this work, we present eight solutions that used deep reinforcement learning approaches, based on algorithms such as Deep Deterministic Policy Gradient, Proximal Policy Optimization, and Trust Region Policy Optimization. Many solutions use similar relaxations and heuristics, such as reward shaping, frame skipping, discretization of the action space, symmetry, and policy blending. However, each of the eight teams implemented different modifications of the known algorithms.

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We present a class of decomposable inequality indices for ordinal data (e.g. self-reported health survey). It is characterized by well-known inequality axioms (e.g. scale invariance) and a decomposability axiom which states that an index can be represented as a function of inequality values in subgroups and subgroup sizes. The only decomposable indices are strictly monotonic transformations of the weighted average of frequencies in categories. Among the indices proposed in the literature only the absolute value index (Abul Naga and Yalcin, 2008; Apouey, 2007) is decomposable. As an empirical illustration we calculate regional contributions to overall health inequality in Switzerland.

The truncated variation, TV c , is a fairly new concept introduced in [5]. Roughly speaking, given a càdlàg function f , its truncated variation is "the total variation which does not pay attention to small changes of f , below some threshold c > 0". The very basic consequence of such approach is that contrary to the total variation, TV c is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in [6], another characterization of TV c was found. Namely TV c is the smallest possible total variation of a function which approximates f uniformly with accuracy c/2. Due to these properties we envisage that TV c might be a useful concept both in the theory and applications of stochastic processes.For this reason we decided to determine some properties of TV c for some well-known processes. In course of our research we discover intimate connections with already known concepts of the stochastic processes theory.Firstly, for semimartingales we proved that TV c is of order c −1 and the normalized truncated variation converges almost surely to the quadratic variation of the semimartingale as c ց 0. Secondly, we studied the rate of this convergence. As this task was much more demanding we narrowed to the class of diffusions (with some mild additional assumptions). We obtained the weak convergence to a so-called Ocone martingale. These results can be viewed as some kind of law of large numbers and the corresponding central limit theorem.Finally, for a Brownian motion with a drift we proved the behavior of TV c on intervals going to infinity. Again, we obtained a LLN and CLT, though in this case they have a different interpretation and were easier to prove.All the results above were obtained in a functional setting, viz. we worked with processes describing the growth of the truncated variation in time. Moreover, in the same respect we also treated two closely related quantities -the so-called upward truncated variation and downward truncated variation.