Limit Theorem For Random Sums Research Articles

On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables

We present upper bounds of the integral $$ {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x $$ for 0 ⩽ l ⩽ 1 + δ, where 0 < δ ⩽ 1, Φ(x) is a standard normal distribution function, and $$ {Z}_N={S}_N/\sqrt{\mathrm{E}{S}_N^2} $$ is the normalized random sum with $$ \mathrm{E}{S}_N^2>0\left({S}_N{X}_1+\dots +{X}_N\right) $$ of centered random variables X1,X2, . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .

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, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular 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 and very convenient and almost universal models for the description of statistical regularities in discrete data. We study analytic properties of GNB distributions. We prove that a GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. We write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, we obtain a representation for the GNB distribution as a mixed geometric distribution. We consider the corresponding scheme of Bernoulli trials with random probability of success. Within this scheme, we prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws. We prove limit theorems 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. We obtain that the class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. We obtain various representations for the limit laws in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. We prove limit theorems establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. We also discuss some applications of GNB distributions in meteorology.

On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables

We present upper bounds for supx ∈ ℝ|P{Z N < x} − Φ(x)|, where Φ(x) is the standard normal distribution function, for random sums $$ {Z}_N={S}_N/\sqrt{\mathbf{V}{S}_N} $$ with variances VS N > 0 (S N = X1 + ⋯ + X N ) of centered strongly mixing or uniformly strongly mixing random variables X1, X2, . . . . Here the number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .

On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N 0 (S N = X 1 + · · · + X N ) of centered independent random variables X 1 ,X 2 , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X 1 ,X 2 , . . . .

Asymptotic inference for non-supercritical partially observed branching processes

To estimate the offspring mean of a branching process one needs observed population sizes up to some generation. However, in applications very often not all individuals existing in the population are observed. Therefore the question about possibility of estimating the population mean based on partial observations is of interest. In existing literature this problem has been studied assuming that the process never becomes extinct, which is possible only in supercritical case. In the paper we consider it in subcritical and critical processes with a large number of initial ancestors. We prove that the Harris type ratio estimator remains consistent, if we have observations of a binomially distributed subsets of the population. To obtain the asymptotic normality of the estimator we modify the estimator using a “skipping” method. The proofs use the law of large numbers and the central limit theorem for random sums in the case when the number of terms and the terms in the sum are not independent.

Weak convergence to the Student and Laplace distributions

Abstract One often observed empirical regularity is a power-law behavior of the tails of some distribution of interest. We propose a limit law for normalized random means that exhibits such heavy tails irrespective of the distribution of the underlying sampling units: the limit is a t-distribution if the random variables have finite variances. The generative scheme is then extended to encompass classic limit theorems for random sums. The resulting unifying framework has wide empirical applicability which we illustrate by considering two empirical regularities in two different fields. First, we turn to urban geography and explain why city-size growth rates are approximately t-distributed, using a model of random sector growth based on the central place theory. Second, turning to an issue in finance, we show that high-frequency stock index returns can be modeled as a generalized asymmetric Laplace process. These empirical illustrations elucidate the situations in which heavy tails can arise.

On asymmetric generalization of the Weibull distribution by scale–location mixing of normal laws

Two approaches are suggested to the definition of asymmetric generalized Weibull distribution. These approaches are based on the representation of the two-sided Weibull distributions as variance–mean normal mixtures or more general scale–location mixtures of the normal laws. Since both of these mixtures can be limit laws in limit theorems for random sums of independent random variables, these approaches can provide additional arguments in favor of asymmetric two-sided Weibull-type models of statistical regularities observed in some problems related to stopped random walks, in particular, in problems of modeling the evolution of financial markets

Asymptotic results for random sums of dependent random variables

Our main result is a central limit theorem for random sums of the form ∑i=1NnXi, where {Xi}i≥1 is a stationary m-dependent process and Nn is a random index independent of {Xi}i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on m-dependent sequences. Further, a weak law of large numbers is proven for ∑i=1NnXi, and the results are exemplified with applications on moving average and descent processes.

L 1 bounds for asymptotic normality of random sums of independent random variables

We present upper bounds of L1 norms in the central limit theorem for random sums of independent random variables X1, X2, . . . , where the number of summands N is a nonnegative integer-valued random variable, independent of the summands X1, X2, . . . .

On the rate of convergence in limit theorems for random sums via Trotter-distance

The main purpose of this paper is to establish some estimates for the rates of convergence in limit theorems for random sums of independent identically distributed random variables via Trotter-distance.MSC:60F05, 60G50, 41A25.

On the Distribution of Crop Yields: Does the Central Limit Theorem Apply?

In this article, we investigate the applicability of the central limit theorem (CLT) to aggregate crop yields. We argue that the aggregation of elementary crop yields is likely to produce nonnormal distributions if, contrary to the standard CLT case, the number of crop acres exhibits substantial time variation. This case is covered by limit theorems for random sums of random variables, which predict nonnormal limiting distributions. The case of substantial variation in the number of summands produces an empirical hypothesis that we test using data from U.S. aggregate state crop yields. The results provide empirical support against the applicability of the CLT.

Almost sure versions of limit theorems for random sums of multiindex random variables

In this paper we obtain an almost sure version of a limit theorem for random sums of multiindex random variables that belong to the domain of attraction of a p-stable law.

An almost sure limit theorem for random sums of independent random variables in the domain of attraction of a semistable law

In this paper we prove an almost sure limit theorem for random sums of independent random variables in the domain of attraction of a p-semistable law and describe the limit law.

Almost sure limit theorems for random sums of multiindex random variables

In this paper, we obtain an almost sure functional limit theorem for random sums of multiindex random variables.

Stochastic foundations of the universal dielectric response

We present a probabilistic model of the microscopic scenario of dielectric relaxation. We prove a limit theorem for random sums of a special type that appear in the model. By means of the theorem, we show that the presented approach to relaxation phenomen

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented.

On the Convergence of Distributions of Random Sums ofIndependent Random Variables to Stable Laws

Previous article Next article On the Convergence of Distributions of Random Sums ofIndependent Random Variables to Stable LawsV. Yu. KorolevV. Yu. Korolevhttps://doi.org/10.1137/S0040585X97976544PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe paper gives the necessary and sufficient conditions for convergence of distributions of random sums of independent random variables to strictly stable laws.[1] Google Scholar[2] V. Yu. Korolev, Convergence of random sequences with independent random indices, Theory Probab. Appl., 39 (1994), pp. 282–297. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkGoogle Scholar[3] Google ScholarKeywordsstable lawsLindeberg conditionrandom sums Previous article Next article FiguresRelatedReferencesCited byDetails Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems8 May 2020 | Mathematics, Vol. 8, No. 5 Cross Ref From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes15 April 2020 Cross Ref Some functional limit theorems for compound Cox processes Cross Ref Application of Compound Cox Processes In Modeling Order Flows with Non-Homogeneous IntensitiesSSRN Electronic Journal Cross Ref Modeling High-Frequency Order Flow Imbalance by Functional Limit Theorems for Two-Sided Risk ProcessesSSRN Electronic Journal Cross Ref On Convergence of Distributions of Compound Cox Processes to Stable LawsV. Yu. Korolev25 July 2006 | Theory of Probability & Its Applications, Vol. 43, No. 4AbstractPDF (119 KB) Volume 42, Issue 4| 1998Theory of Probability & Its Applications History Published online:17 February 2012 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordsstable lawsLindeberg conditionrandom sumsPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97976544Article page range:pp. 695-696ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Estimates for the Convergence Rate to the Poisson Distribution for Random Sums of Independent Indicators

Previous article Next article Estimates for the Convergence Rate to the Poisson Distribution for Random Sums of Independent IndicatorsP. L. LogunovP. L. Logunovhttps://doi.org/10.1137/1135083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. Rakhimov, A limit theorem for random sums of dependent indicators and its applications in the theory of branching processes, Theory Probab. Appl., 32 (1987), 290–298 10.1137/1132038 0663.60025 LinkGoogle Scholar[2] A. M. Zubkov, Inequalities for the distributions of sums of functions of independent random variables, Math. Notes, 22 (1987), 745–758, (In Russian.) Google Scholar[3] V. M. Zolotarev, The Modern Theory of Summation of Independent Random Variables, Nauka, Moscow, 1986, (In Russian.) 0649.60016 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Poisson approximationProbability Surveys, Vol. 16, No. none Cross Ref On the distance between the distributions of random sums14 July 2016 | Journal of Applied Probability, Vol. 40, No. 1 Cross Ref Poisson and compound Poisson approximations for random sums of random variables14 July 2016 | Journal of Applied Probability, Vol. 33, No. 1 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications History Submitted:30 May 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135083Article page range:pp. 587-590ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Addendum: A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes

Addendum: A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes

A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes

A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes