Independent Random Variables In Terms Research Articles

EM Analysis of Geometrical Uncertainty for Metallic/Dielectric Bodies of Revolution Targets

An efficient EM scattering solution of geometrical uncertainty is fabricated to analyze the scattering from electrically large metallic/dielectric bodies of revolution (BoRs). The algorithm started by describing the geometrical shape of the BoR with the two-dimensional non-uniform rational B-spline curves. Then the relationship can be constructed between the local random variables and the spatial BoR basis functions. Thus, the surface integral equations were represented in terms of independent random variables. Radar cross section can be quickly obtained for perturbed geometrical shapes and the proposed algorithm can save computational resources when compared with theMonte Carlo (MC) method. The electrical field integral equation (EFIE) is applied to calculate the metallic BoR targets, whereas the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation is used to analyze the dielectric BoR targets. Numerical results are given to demonstrate the accuracy and efficiency of the proposed method.

Moments of the Scores

Upper bounds on absolute moments of the scores are derived for sums of independent random variables in terms of the moments of the scores, as well as in terms of the total variation norm of densities of summands.

On Capacities of the Two-User Union Channel With Complete Feedback

The exact values of the optimal symmetric rate point in the Cover--Leung capacity region of the two-user union channel with complete feedback were determined by Willems when the size of the input alphabet is 2, and by Vinck, Hoeks and Post when the size is at least 6. We complete this line of research when the size of the input alphabet is 3, 4 or 5. The proof hinges on the technical lemma that concerns the maximal joint entropy of two independent random variables in terms of their probability of equality. For the zero-error capacity region, using superposition coding, we provide a practical near-optimal communication scheme which improves all the previous explicit constructions.

A Generalization of the Rozovskii Inequality

Adopting ideas of Katz (1963), Petrov (1965), Wang and Ahmad (2016), and Gabdullin, Makarenko, and Shevtsova (2016), we generalize the Rozovskii inequality (1974) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms of the absolute value of the sum of truncated in a fixed point third-order moments and the sum of the second-order tails of random summands. The generalization is due to introduction of a truncation parameter and a weighting function from a set of functions originally introduced by Katz (1963). The obtained inequality does not assume finiteness of moments of random summands of order higher than the second and may be even sharper than the celebrated inequalities of Berry (1941), Esseen (1942, 1969), Katz (1963), Petrov (1965), and Wang & Ahmad (2016).

A Generalization of the Wang–Ahmad Inequality

By introducing a truncation parameter, we generalize the Ahmad–Wang inequality (2016) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms of weighted absolute values of truncated third-order moments and tails of the second-order moments of random summands. The obtained estimate also generalizes the celebrated inequalities due to Berry (1941), Esseen (1942, 1969), Katz (1963), and Petrov (1965).

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in dd-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Ito stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.

Estimating Averages of Order Statistics of Bivariate Functions

We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for the expected value of averages of order statistics of sequences of independent random variables in terms of Orlicz norms are obtained. In the case where the bivariate functions are matrices, we provide a "minimal" probability space which allows us to $C$-embed certain Orlicz spaces $\ell_M^n$ into $\ell_1^{cn^3}$, $c,C>0$ being absolute constants.

Exponential inequality for a class of NOD random variables and its application

In this paper, an exponential inequality for weighted sums of identically distributed NOD (negatively orthant dependent) random variables is established, by which we obtain the almost sure convergence rate O(1)n−1/2(log n)1/2 of \( \sum\limits_{i = 1}^n {a_{ni} {{\left( {X_i - EX_i } \right)} \mathord{\left/ {\vphantom {{\left( {X_i - EX_i } \right)} n}} \right. \kern-\nulldelimiterspace} n}} \), which reaches the available one for independent random variables in terms of Berstein type inequality. As application, we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples, from which the strong consistency rate O(1)n−1/2hn−1(log n)1/2 s also obtained.

Approximation by the Dickman Distribution and Quasi-Logarithmic Combinatorial Structures

Quasi-logarithmic combinatorial structures are a class of decomposable combinatorial structures which extend the logarithmic class considered by Arratia, Barbour and Tavaré (2003). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. This in turn requires an argument that refines the Mineka coupling by incorporating a blocking construction, leading to exponentially sharper coupling rates for the sums in question. Applications include distributional limit theorems for the size of the largest component and for the vector of counts of the small components in a quasi-logarithmic combinatorial structure.

On the strong convergence rate for positively associated random variables

For sequences of the positively associated random variables which are strictly weaker than the classical associated random ones introduced by Esary et al. (1967) [7], strong convergence rate n − 1 / 2 ( log n ) 1 / 2 is obtained, which reaches the available one for independent random variables in terms of Berstein type inequality. Further, we give the corresponding precise asymptotics with respect to the rate mentioned above, which extend and improve the relevant results in Fu (2009) [8].

A stochastic Lagrangian approach for geometrical uncertainties in electrostatics

This work proposes a general framework to quantify uncertainty arising from geometrical variations in the electrostatic analysis. The uncertainty associated with geometry is modeled as a random field which is first expanded using either polynomial chaos or Karhunen–Loève expansion in terms of independent random variables. The random field is then treated as a random displacement applied to the conductors defined by the mean geometry, to derive the stochastic Lagrangian boundary integral equation. The surface charge density is modeled as a random field, and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method, respectively. Various numerical examples are presented to study the effect of uncertain geometry on relevant parameters such as capacitance and net electrostatic force. The results obtained using the proposed method are verified using rigorous Monte Carlo simulations. It has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters.

On the Stein–Tikhomirov method and its applications in nonclassical limit theorems

In this paper we give necessary and sufficient conditions for validity of the central limit theorem for sums of independent random variables in terms of characteristic functions in a nonclassical setup where an essential contribution of individual variables to the sum is allowed. These conditions are formulated with the use of the Stein–Tikhomirov operator.

A probabilistic approach to the geometry of the ℓpn-ball

This article investigates, by probabilistic methods, various geometric questions on B n p , the unit ball of ln p . We propose realizations in terms of independent random variables of several distributions on B n p , including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B n p . As another application, we compute moments of linear functionals on B n p , which gives sharp constants in Khinchine's inequalities on B n p and determines the 2-constant of all directions on B n p . We also study the extremal values of several Gaussian averages on sections of B n p (including mean width and l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition.

Lower Bounds for Probabilities of Large Deviations of Sums of Independent Random Variables

We derive lower bounds for probabilities of large deviations of sums of independent random variables in terms of tail probabilities for the number of successes in nonhomogeneous Bernoulli trials. These bounds are convenient if the Lyapunov ratio is great, and also in the case of bounded summands.

Probabilistic Inequalities for Sums of Independent RandomVariables in Terms of Truncated Pseudomoments

An estimate is obtained for the probability of large deviations of the sum of independent random variables which uses the closeness of the distributions of summands to the Gaussian distribution.}

A characterization of probability distributions similar to the exponential

AbstractA concept of the lack‐of‐memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost‐lack‐of‐memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed.

Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large Deviations

Previous article Next article Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large DeviationsS. Ya. ShorginS. Ya. Shorginhttps://doi.org/10.1137/1127034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Sazonov, A new general estimate of the rate of convergence in the central limit theorem in $R^{k}$, Proc. Nat. Acad. Sci. USA, 71 (1974), 118–121 0277.60010 CrossrefGoogle Scholar[2] V. I. Rotar', Non-classical estimates of the rate of convergence in the multi-dimensional central limit theorem. I, Theory Prob. Appl., 22 (1977), 755–772 0391.60026 LinkGoogle Scholar[3] V. I. Rotar', Non-classical estimates for the rate of convergence in the multidimensional central limit theorem. II, Theory Prob. Appl., 23 (1978), 50–62 0423.60021 LinkGoogle Scholar[4] S. V. Nagaev and , V. I. Rotar', Sharpening of Lyapunov type estimates (the case when the distributions of the summands are close to the normal distribution), Theory Prob. Appl., 18 (1973), 107–119 LinkGoogle Scholar[5] S. Ya. Shorgin, A non-classical estimate of the rate of convergence in the multi-dimensional central limit theorem taking into account large deviations, Theory Prob. Appl., 23 (1978), 667–671 LinkGoogle Scholar[6] S. V. Nagaev and , S. K. Sakoyan, An estimate for the probability of large deviations, Limit theorems and mathematical statistics (Russian), Izdat. “Fan” Uzbek. SSR, Tashkent, 1976, 132–140, 190 55:9237 Google Scholar[7] S. K. Sakoyan, Estimates for distribution functions of sums of random variables taking into account large deviations, 1977, author's abstract of dissertation, Tashkent Google Scholar[8] V. I. Rotar', Masters Thesis, Limit Theorems for Linear and Multilinear Forms, dissertation, LGU, Moscow-Leningrad, 1978 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Probabilistic Inequalities for Sums of Independent RandomVariables in Terms of Truncated PseudomomentsS. V. NagaevTheory of Probability & Its Applications, Vol. 42, No. 3 | 17 February 2012AbstractPDF (265 KB)On the Exit of a Random Walk From a Curvilinear BoundaryM. U. Gafurov and V. I. Rotar’Theory of Probability & Its Applications, Vol. 28, No. 1 | 17 July 2006AbstractPDF (477 KB) Volume 27, Issue 2| 1983Theory of Probability & Its Applications215-440 History Submitted:26 March 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127034Article page range:pp. 324-337ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Large Deviations of Sums of Independent Random Variables

This paper deals with numerous variants of bounds for probabilities of large deviations of sums of independent random variables in terms of ordinary and generalized moments of individual summands. A great deal of attention is devoted to the study of the precision of these bounds. In this connection comparisons are made with precise asymptotic results. At the end of the paper various applications of the bounds for probabilities of large deviations to the strong law of large numbers, the central limit theorem and to certain other problems are discussed.

Conservative processes with stochastic rates

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

Conservative processes with stochastic rates

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.