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In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments Lévy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving Lévy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three different cases. Our work is related to [9,10], who considered power variation functionals of stationary increments Lévy driven moving averages.

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Rank histograms are popular tools for assessing the reliability of meteorological ensemble forecast systems. A reliable forecast system leads to a uniform rank histogram, and deviations from uniformity can indicate miscalibrations. However, the ability to identify such deviations by visual inspection of rank histogram plots crucially depends on the number of bins chosen for the histogram. If too few bins are chosen, the rank histogram is likely to miss miscalibrations; if too many are chosen, even perfectly calibrated forecast systems can yield rank histograms that do not appear uniform. In this paper we address this trade‐off and propose a method for choosing the number of bins for a rank histogram. The goal of our method is to select a number of bins such that the intuitive decision whether a histogram is uniform or not is as close as possible to a formal statistical test. Our results indicate that it is often appropriate to choose fewer bins than the usual choice of ensemble size plus one, especially when the number of observations available for verification is small.

In this paper we present some limit theorems for power variation of Lévy semistationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in [14], where the authors derived the limit theory for kth order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power p > 0, the Blumenthal-Getoor index β ∈ (0, 2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α. In this paper we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.