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It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples., Comment: Important and necessary changes have been made in this new version, this version supersedes version 1

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property., Comment: 21 pages, no figures

We prove a second Marshall inequality for adaptive convex density estimation via least squares. The result completes the first inequality proved recently by Dumbgen et al. [2007. Marshall's lemma for convex density estimation. IMS Lecture Notes—Monograph Series, submitted for publication. Preprint available at 〈 http://arxiv.org/abs/math.ST/0609277 〉 ], and is very similar to the original Marshall inequality in monotone estimation.