Related papers: Multivariate $\alpha$-normal distributions
A well-known result across information theory, machine learning, and statistical physics shows that the maximum entropy distribution under a mean constraint has an exponential form called the Gibbs-Boltzmann distribution. This is used for…
Using a recently derived integral in terms of elementary functions, we derive new asymptotic expansions of the normal inverse Gaussian cumulative distribution function. One of the asymptotic representations is in terms of the normal…
This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the $n$ partitions of the interval $[0,W_n]$ are independent and identically distributed…
Some probability distributions have moments, and some do not. For example, the normal distribution has power moments of arbitrary order, but the Cauchy distribution does not have power moments. In this paper, by analogy with the…
We find a general formula for the distribution of time-averaged observables for systems modeled according to the sub-diffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and…
In this paper, asymptotic expansions of the distributions and densities of powered extremes for Maxwell samples are considered. The results show that the convergence speeds of normalized partial maxima relies on the powered index.…
We generalize the usual exponential Boltzmann factor to any reasonable and potentially observable distribution function, $B(E)$. By defining generalized logarithms $\Lambda$ as inverses of these distribution functions, we are led to a…
A five-parameter distribution called the McDonald normal distribution is defined and studied. The new distribution contains, as special cases, several important distributions discussed in the literature, such as the normal, skew-normal,…
In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show,…
We show that whenever data are gathered using a device that performs a normalization-preprocessing, the ensuing normalized input, as recorded by the measurement device, will always be q-Gaussian distributed if the incoming data exhibit…
We calculate the single-particle momentum distribution of a quantum many-particle system in the presence of the Coulomb interaction and a confining potential. The region of intermediate momenta, where the confining potential dominates,…
The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar non-equilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained…
The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation…
We study the probability distribution of stock returns at mesoscopic time lags (return horizons) ranging from about an hour to about a month. While at shorter microscopic time lags the distribution has power-law tails, for mesoscopic times…
The generalized inverse Gaussian-Poisson (GIGP) distribution proposed by Sichel in the 1970s has proved to be a flexible fitting tool for diverse frequency data, collectively described using the item production model. In this paper, we…
A random variable is equi-dispersed if its mean equals its variance. A Poisson distribution is a classical example of this phenomenon. However, a less well-known fact is that the class of normal densities that are equi-dispersed constitutes…
Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of…
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue…
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular,…
A practical statistical analysis on the regional populations and GDPs of China is conducted. The result shows that the distribution of the populations and that of the GDPs obeys the shifted power law, respectively. To understand these…