3,027 research outputs found
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
Histogram Arithmetic under Uncertainty of Probability Density Function
In this article we propose a method of performing arithmetic operations on
varia-bles with unknown distribution. The approach to the evaluation results of
arithme-tic operations can select probability intervals of the algebraic
equations and their systems solutions, of differential equations and their
systems in case of histogram evaluation of the empirical density distributions
of random parameters.Comment: 10 page
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 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
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 -particles
from -sources of U deposited on the chamber electrodes was used
for absolute normalization of the W values. The energy dependence of for
protons in H shows an anomalous increase of 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
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