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A class of estimators of the Rényi and Tsallis entropies of an unknown distribution f in R m is presented. These estimators are based on the kth nearest-neighbor distances computed from a sample of N i.i.d. vectors with distribution f . We show that entropies of any order q, including Shannon's entropy, can be estimated consistently with minimal assumptions on f . Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each.

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Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.

Pearson diffusions are governed by diffusion equations with polynomial coefficients. Fractional Pearson diffusions are governed by the corresponding time-fractional diffusion equation. They are useful for modeling sub-diffusive phenomena, caused by particle sticking and trapping. This paper provides explicit strong solutions for fractional Pearson diffusions, using spectral methods. It also presents stochastic solutions, using a non-Markovian inverse stable time change.

Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented. Sojourn measures of Student and Fisher-Snedecor random fields3 The paper has three aims. One is to provide explicit, albeit asymptotic, formulae for the distribution of the volume of excursion sets of a class of strongly dependent random fields. The second one is to derive asymptotic results for heavy-tailed random fields. Finally, the third aim is to generalize the previous findings to sojourn measures above moving levels and for cross-correlated scenarios.There is, therefore, a need for models that are able to display strongly dependent heavytailed behaviour and yet are sufficiently simple to allow analysis. To obtain explicit results we detail the underlying structure of random fields. Namely, a basic assumption of the analysis is that we examine functionals of vector Gaussian random fields, in particular, Student and Fisher-Snedecor random fields. Consult [3,15,16,47] on excursion sets of chi-square, Student and Fisher-Snedecor random fields and their importance for image analysis and studies of brain function. Other results on sojourn measures of chi-square random fields can be found in [23,27,29,30].Minkowski functionals are widely used to characterise geometric properties of random fields, in particular in the analysis of cosmic microwave background radiation, see [36,38]. In this paper we investigate the first Minkowski functional of random fields and its expansions into multidimensional Hermite polynomials, see some one-dimensional/discrete counterparts in [18,20]. To have a complete account of results on asymptotic distributions of sojourn measures for functions of vector random fields, we also prove corresponding theorems for weakly dependent scenarios.The remainder of the paper is structured as follows. In Sections 2-4, we introduce the necessary background from the theory of random fields and briefly review some definitions and notation on the first Minkowski functional, multidimensional Hermite expansions, and Student and Fisher-Snedecor random fields. We start Sections 5 and 7 with generalizations and corrections of some classical asymptotic results to arbitrary sets and vector fields. With this in hand, we continue Sections 5 and 7 by new results for the first Minkowski functional of Student and Fisher-Snedecor random fields. In Section 7, we also show how to lift these results to sojourn measures above moving levels and for cross-correlated underlying vector fields. Sections 6 and 8 provide the proofs of all theorems and lemmata in the article. Simulation results on the limit distributions of areas of excursion sets for two types of images are given in Section 9. Short conclusions are made in Section 10.In this paper, we only consider real-valued random f...