Локальные предельные теоремы для сложных дискретных распределений

A local limit theorem for hidden Markov chains

Let X1, X2, · ··, Xn, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X1, · ··, Xm) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

A large deviation limit theorem for multivariate distributions

Missing values are a problem that is often encountered in various fields and must be addressed to obtain good statistical inference such as parameter estimation. Missing values can be found in any type of data, included count data that has Poisson distributed. One solution to overcome that problem is applying multiple imputation techniques. The multiple imputation technique for the case of count data consists of three main stages, namely the imputation, the analysis, and pooling parameter. The use of the normal distribution refers to the sampling distribution using the central limit theorem for discrete distributions. This study is also equipped with numerical simulations which aim to compare accuracy based on the resulting bias value. Based on the study, the solutions proposed to overcome the missing values in the count data yield satisfactory results. This is indicated by the size of the bias parameter estimate is small. But the bias value tends to increase with increasing percentage of observation of missing values and when the parameter values are small.

Because of digitalization, many organizations possess large datasets. Furthermore, measurement data are often not normally distributed. However, when samples are sufficiently large, the central limit theorem may be used for the sample means. In this article, we evaluate the use of the central limit theorem for various distributions and sample sizes, as well as its effects on the performance of a Shewhart control chart for these large non‐normally distributed datasets. To this end, we use the sample means as individual observations and a Shewhart control chart for individual observations to monitor processes. We study the unconditional performance, expressed as the expectation of the in‐control average run length (ARL), as well as the conditional performance, expressed as the probability that the control chart based on estimated parameters will have a lower in‐control ARL than a specified desired in‐control ARL. We use recently developed factors to correct the control limits to obtain a specified conditional or unconditional in‐control performance. The results in this paper indicate that the control chart should be applied with caution, even with large sample sizes.

Complete monotonicity of the entropy in the central limit theorem for gamma and inverse Gaussian distributions

The aim of this note is to discuss some statistical properties of the phase separation line in the 2D low-temperature Ising model. We prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation. The limiting Gaussian measure corresponds to a scaled Brownian bridge with direction dependent parameters. Up to the temperature factor, the variances of local increments of this limiting process are inversely proportional to the stiffness.

The Central Limit Theorem (CLT) plays a foundational role in statistical inference, often serving as the rationale for assuming a normal approximation of the sample mean. Yet, the pace at which this assumption becomes valid is influenced by the shape of the parent distribution, especially its skewness. This research quantifies the minimum number of observations required for the mean of samples drawn from skewed, non-normal distributions specifically Gamma, Poisson, Binomial, and Beta to achieve a satisfactory normal approximation. We implemented a Monte Carlo simulation and applied both the Shapiro-Wilk and Kolmogorov-Smirnov tests to assess the adequacy of the normal approximation. Results indicate a nonlinear association between the degree of skewness and the sample size required for acceptable normal approximation. For distributions with mild asymmetry (|skewness| < 0.5), 20 samples often suffice, whereas more heavily skewed distributions (|skewness| ≥ 2.5) may necessitate sample sizes beyond 100. These findings call into question the blanket use of the "n ≥ 30" heuristic and suggest more tailored guidelines are necessary for accurate inference. A graphical overview summarizes these results across the examined distributional families, offering clear guidance for applied researchers working with non-normal data.

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

For a positive integer N N , let X N {X_N} be a random variable uniformly distributed over the set { log ⁡ d : d | N } \{ \log d:d|N\} . Let F N {F_N} be the normalized (to have expectation zero and variance one) distribution function for X N {X_N} . Necessary and sufficient conditions for the convergence of a sequence F N j {F_{{N_j}}} of distributions are given. The possible limit distributions are investigated, and the case where the limit distribution is normal is considered in detail.

For every N ≧ 1, let be a real-valued stationary process, and let . Suppose that and var, where ∊N and τN are positive null sequences. Limiting distributions of as N → ∞ are obtained for the cases τN = ∊N and τN = o(∊N ). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of applications to genetic models.

Let { F n} be a filtration, { X n } an adapted sequence of real random variables, and { α n } a predictable sequence of non-negative random variables with α 1>0. Set β n= ∑ i=1 n α i and define the random distribution functions F n(t)=(1/β n) ∑ i=1 n α iI {X i⩽t} and B n(t)=(1/β n) ∑ i=1 n α iP(X i⩽t| F i−1) . Under mild assumptions on { α n }, it is shown that sup t |F n(t)−B n(t)|→0 , a.s. on the set { F n or B n converges uniformly } . Moreover, conditions are given under which F n converges uniformly with probability 1.

Previous article Next article On a Lower Bound for the Convergence Rate in a Local Limit TheoremV. K. MatskyavichyusV. K. Matskyavichyushttps://doi.org/10.1137/1130100PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] William Feller, An introduction to probability theory and its applications. Vol. II. , Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 42:5292 0219.60003 Google Scholar[2] A. P. Prudnikov, , Yu. A. Brychkov and , O. I. Marichev, Integrals and Series: Elementary Functions, Nauka, Moscow, 1981, (In Russian.) 0511.00044 Google Scholar[3] M. Abramowitz and , I. A. Stegun, Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966xiv+1046 34:8607 CrossrefGoogle Scholar[4] V. K. Matskyavichyus, A lower bound for the convergence rate in the central limit theorem, Theory Prob. Appl., 28 (1983), 596–601 0543.60027 LinkGoogle Scholar[5] V. K. Matskyavichyus, Letters to the editor, Theory Prob. Appl., 29 (1984), 196–197 LinkGoogle Scholar[6] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346 52:9335 0322.60042 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Approximate local limit theorems with effective rate and application to random walks in random sceneryBernoulli, Vol. 23, No. 4B Cross Ref Local limit theorems in some random models from number theory1 September 2016 | Stochastic Analysis and Applications, Vol. 34, No. 6 Cross Ref Volume 30, Issue 4| 1986Theory of Probability & Its Applications History Submitted:05 August 1985Published online:28 July 2006 InformationCopyright © 1986 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130100Article page range:pp. 810-814ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^{d}$, $d\geq 2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the $k$-face and intrinsic volume functionals.

Local limit theorems are derived for sums of finite range $\mathbb{Z}$-valued potential functions of an iid random field. The resulting approximations turn out to be mixtures of standard normal densities for lattice distributions supported by residue classes of integers. The mixing weights are equal to the probability that the sum of potential functions lies in such a residue class and are nonasymptotic and computable. For finite range potential functions of a stationary Gibbsian random field with bounded and finite range interactions, conditions are given under which the global central limit theorem implies the classical local limit theorem.