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Stochastic dominance of a random variable by a convex combination of its independent copies has recently been shown to hold within the relatively narrow class of distributions with concave odds function, and later extended to broader…

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding…

We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

We discuss non-Gaussian random matrices whose elements are random variables with heavy-tailed probability distributions. In probability theory heavy tails of the distributions describe rare but violent events which usually have dominant…

Beck et al. (2013) introduced a new distribution class J which contains many heavy-tailed and light-tailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavy-tailed distributions to…

Some new survival distributions are introduced based on a generalised exponential function. This class of distributions includes heavy-tailed generalisations of exponential, Weibull and gamma distributions. Properties of the distributions…

We propose the notion of sub-Weibull distributions, which are characterised by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalises the sub-Gaussian and sub-Exponential families to…

The random graph model has recently been extended to a random preferential attachment graph model, in order to enable the study of general asymptotic properties in network types that are better represented by the preferential attachment…

This paper considers learning deep features from long-tailed data. We observe that in the deep feature space, the head classes and the tail classes present different distribution patterns. The head classes have a relatively large spatial…

Real-world data are long-tailed, the lack of tail samples leads to a significant limitation in the generalization ability of the model. Although numerous approaches of class re-balancing perform well for moderate class imbalance problems,…

In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. [1]. We describe the structure of this class from different…

Skew-elliptical distributions constitute a large class of multivariate distributions that account for both skewness and a variety of tail properties. This class has simpler representations in terms of densities rather than cumulative…

In this paper, we give some conditions, under which, if an infinitely divisible distribution supported on $[0,\infty)$ belongs to the intersection of exponential distribution class $\mathcal{L}(\gamma)$ for some $\gamma\ge0$ and generalised…

Different questions related with analysis of extreme values and outliers arise frequently in practice. To exclude extremal observations and outliers is not a good decision because they contain important information about the observed…

We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Levy trees. This extends previous results due to several authors.

We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not…

The task for a general and useful classification of the tail behaviors of probability distributions still has no satisfactory solution. Due to lack of information outside the range of the data the tails of the distribution should be…

We study large deviation properties of probability distributions with either a compact support or a fat tail by comparing them with q-deformed exponential distributions. Our main result is a large deviation property for probability…

We characterise the learning of a mixture of two clouds of data points with generic centroids via empirical risk minimisation in the high dimensional regime, under the assumptions of generic convex loss and convex regularisation. Each cloud…

We investigate a way of comparing and classifying tails of random variables. Our approach extends the notion of classical indices, such as exponential and moment indices, which are widely used measuring heaviness of tail functions. A…