Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions

Bounds of the accuracy of the normal approximation to the distribution of a sum of independent random variables are improved under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. These results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same conditions, bounds are obtained for the accuracy of the approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, these bounds are constructed for the accuracy of approximation of the distributions of geometric, negative binomial and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma and Student distributions, respectively. All absolute constants are written out explicitly.Comment: 20 pages, research supported by the Russian Foundation of Basic Research, project 15-07-0298

On the asymptotic approximation to the probability distribution of extremal precipitation

Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.Comment: 17 pages, 6 figures. arXiv admin note: text overlap with arXiv:1703.0727

Convergence of random sums and statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions and their generalizations

We present some product representations for random variables with the Linnik, Mittag-Leffler and Weibull distributions and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the scheme of geometric summation is far not the only asymptotic setting (even for sums of independent random variables) in which the Mittag-Leffler and Linnik laws appear as limit distributions. The two-sided Mittag-Leffler and one-sided Linnik distribution are introduced and also proved to be limit laws for some statistics constructed from samples with random sizes

Generalized negative binomial distributions as mixed geometric laws and related limit theorems

In this paper we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical regularities in discrete data. Analytic properties of GNB distributions are studied. A GG distribution is proved to be a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GNB distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. The class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Some applications of GNB distributions in meteorology are discussed

Sedeonic theory of massless fields

In present paper we develop the description of massless fields on the basis of space-time algebra of sixteen-component sedeons. The generalized sedeonic second-order equation for unified gravitoelectromagnetic (GE) field describing simultaneously gravity and electromagnetism is proposed. The second-order relations for the GE field energy, momentum and Lorentz invariants are derived. We consider also the generalized sedeonic first-order equation for the massless neutrino field. The second-order relations for the neutrino potentials analogues to the Pointing theorem and Lorentz invariant relations in gravitoelectromagnetism are also derived.Comment: 18 page

Modeling high-frequency order flow imbalance by functional limit theorems for two-sided risk processes

A micro-scale model is proposed for the evolution of the limit order book. Within this model, the flows of orders (claims) are described by doubly stochastic Poisson processes taking account of the stochastic character of intensities of bid and ask orders that determine the price discovery mechanism in financial markets. The process of {\it order flow imbalance} (OFI) is studied. This process is a sensitive indicator of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. The OFI process tracks best bid and ask queues and change much faster than prices. It incorporates information about build-ups and depletions of order queues so that it can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The {\it two-sided risk processes} are suggested as mathematical models of the OFI process

On mixture representations for the generalized Linnik distribution and their applications in limit theorems

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1602.02480, arXiv:1506.0277

A note on functional limit theorems for compound Cox processes

An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to L{\'e}vy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to L{\'e}vy processes with variance-mean mixed normal distributions, in particular, to stable L{\'e}vy processes, generalized hyperbolic and generalized variance-gamma L{\'e}vy processes.Comment: arXiv admin note: substantial text overlap with arXiv:1410.190

Energy dependence of W values for protons in hydrogen

The mean energy $W$ required to produce an ion pair in molecular hydrogen has been obtained for protons in the energy range between 1 MeV and 4.5 MeV. The W values were derived from the existing experimental data on elastic {\it $\pi^-$p} scattering at the beam energy of 40 GeV. In the experiment, the ionization chamber IKAR filled with hydrogen at a pressure of 10 at served simultaneously as a gas target and a detector for recoil protons. For selected events of elastic scattering, the ionization yield produced by recoil protons was measured in IKAR, while the energy was determined kinematically through the scattering angles of the incident particles measured with a system of multi-wire proportional chambers. The ionization produced by $\alpha$-particles from $\alpha$-sources of $^{234}$U deposited on the chamber electrodes was used for absolute normalization of the W values. The energy dependence of $W$ for protons in H$_2$ shows an anomalous increase of $W$ with increasing energy in the measured energy range. At the energy of 4.76 MeV, the ionization yield for alpha particles is by 2\% larger than that for protons.Comment: 12 pages, 5 figure

Statistical tests for extreme precipitation volumes

The approaches, based on the negative binomial model for the distribution of duration of the wet periods measured in days, are proposed to the definition of extreme precipitation. This model demonstrates excellent fit with real data and provides a theoretical base for the determination of asymptotic approximations to the distributions of the maximum daily precipitation volume within a wet period as well as the total precipitation volume over a wet period. The first approach to the definition (and determination) of extreme precipitation is based on the tempered Snedecor-Fisher distribution of the maximum daily precipitation. According to this approach, a daily precipitation volume is considered to be extreme, if it exceeds a certain (pre-defined) quantile of the tempered Snedecor--Fisher distribution. The second approach is based on that the total precipitation volume for a wet period has the gamma distribution. Hence, the hypothesis that the total precipitation volume during a certain wet period is extremely large can be formulated as the homogeneity hypothesis of a sample from the gamma distribution. Two equivalent tests are proposed for testing this hypothesis. Both of these tests deal with the relative contribution of the total precipitation volume for a wet period to the considered set (sample) of successive wet periods. Within the second approach it is possible to introduce the notions of relatively and absolutely extreme precipitation volumes. The results of the application of these tests to real data are presented yielding the conclusion that the intensity of wet periods with extreme large precipitation volume increases.Comment: 21 pages, 10 figures, 2 table