Sums Of Independent Random Variables Research Articles

A remark on the law of the logarithm for weighted sums of random variables with multidimensional indices

In this work, a sharp upper bound on the law of the logarithm for the weighted sums of random variables with multidimensional indices is obtained. The main result improves the result in [Li, Rao and Wang, 1995. On strong law of large numbers and the law of the logarithm for weighted sums of independent random variables with multidimensional indices. J. Multivariate Anal. 52, 181–198], partly.

On the real accuracy of approximation in the central limit theorem

We consider various approximations in the central limit theorem for distributions of sums of independent random variables. We study how many summands in the normalized sums guarantee the precision 10−3 for these approximations. It turns out that for the same distribution but different approximations this number varies from hundreds of thousands to a few tens.

On the limit behavior of increments of sums of independent random variables from domains of attraction of asymmetric stable distributions

The asymptotic behavior of increments of sums of independent identically distributed random variables with incremental length (logn)p is considered. The laws describing increments of such length are intermediate between the Csogő-Revesz law (for large incremental lengths) and the Erdo-Renyi law (for small incremental lengths). A new result for random variables from the domain of normal attraction of asymmetric stable laws with parameter α e (1, 2) is obtained.

Strong convergence on weakly logarithmic combinatorial assemblies

We deal with the random combinatorial structures called assemblies. Instead of the traditional logarithmic condition which assures asymptotic regularity of the number of components of a given order, we assume only lower and upper bounds of this number. Using the author’s analytic approach, we generalize the independent process approximation in the total variation distance of the component structure of an assembly. To evaluate the influence of strongly dependent large components, we obtain estimates of the appropriate conditional probabilities by unconditioned ones. The estimates are applied to examine additive functions defined on a new class of structures, called weakly logarithmic. Some analogs of Major’s and Feller’s theorems which concern almost sure behavior of sums of independent random variables are proved.

Sums of Independent Random Variables of the Modified Makeham Distribution

This paper shows how the renewal process in which the sums of independent random variables of the modified Makeham distribution appear can be approximated by the surrogate distribution.

Peakedness for weighted sums of symmetric random variables

In this paper, we provide sufficient conditions on weighted coefficients under which the peakedness comparison between weighted sums of independent random variables can be carried out. These results extend and enrich the existing peakedness results in the literature including those presented by Proschan (1965) and Ma (1998).

Product-Convolution of R-Symmetric Unimodal Distributions: An Analogue of Wintner's Theorem

Abstract Gnedenko and Kolmogorov (1949) in their acclaimed monograph. Limit Distributions for Sums of Independent Random Variables, claimed that the convolutions of unimodal distributions are unimodal. Kai Lai Chung, in an appendix of his English translation of the monograph, by a counterexample, refuted the claim and further noted Wintner's (1938) result that the convolutions of symmetric unimodal distributions are symmetric unimodal. In this note, it is shown that the product-convolutions of unimodal distributions are not unimodal either. Furthermore, an analogue of Wintner's result based on the relatively recent notion of R-symmetry (Mudholkar and Wang, 2007) is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.

Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model

We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang, and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91–100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev, Probability for sums of independent random variables, Theory Probab. Appl., 16:600–675, 1971].

A new approach to strong embeddings

We revisit strong approximation theory from a new perspective, culminating in a proof of the Komlos–Major–Tusnady embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein’s method of normal approximation, is applicable to any setting where Stein’s method works. In particular, one can hope to take it beyond sums of independent random variables.

Interactive MATLAB‐based demo program for sum of independent random variables

AbstractWe present an interactive MATLAB‐based demo program devoted to teaching undergraduate students the behavior of the sum of independent random variables. The user chooses the type of the variables, and the parameters of these variables. The sum is shown in the time domain followed by the estimation of the probability density of the sum using a histogram. It is subsequently shown that the probability density of the sum of random variables represents the convolution of the density functions of the corresponding variables. The Central Limit Theorem and the progression of the sum towards this limit are illustrated in terms of the number of summands. The software can be used as a complement to theoretical classes or alone as a self‐study tool. Student evaluation of the program is also included. © 2010 Wiley Periodicals, Inc. Comput Appl Eng Educ 21: 372–385, 2013

On Some Densities in the Set of Permutations

The asymptotic density of random permutations with given properties of the $k$th shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables.

Tunstall Code, Khodak Variations, and Random Walks

A variable-to-fixed length encoder partitions the source string into variable-length phrases that belong to a given and fixed dictionary. Tunstall, and independently Khodak, designed variable-to-fixed length codes for memoryless sources that are optimal under certain constraints. In this paper, we study the Tunstall and Khodak codes using variety of techniques ranging from stopping times for sums of independent random variables to Tauberian theorems and Mellin transform. After proposing an algebraic characterization of the Tunstall and Khodak codes, we present new results on the variance and a central limit theorem for dictionary phrase lengths. This analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates.

Theorems on convergence of stochastic integrals distributions to signed measures and local limit theorems for large deviations

We study properties of symmetric stable measures with index α > 2, α ≠ 2m, $$ m \in \mathbb{N} $$ . Such measures are signed ones, and hence they are not probability measures. For this class of measures, we construct an analogue of the Levy–Khinchin representation. We show that, in some sense, these signed measures are limit measures for sums of independent random variables. Bibliography: 11 titles.

On asymptotic behavior of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for iterated processes are quite different. When the iterated process is a homogeneous process with independent increments in which time is replaced by random one, the behavior of large and moderate deviations is studied in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramer condition, the Linnik condition, and the existence of moment of order p > 2 for a positive part. Bibliography: 6 titles.

On sums of independent random variables whose distributions belong to the max domain of attraction of max stable laws

In this article we study the max domains of attraction of distributions of sums of independent random variables belonging to the max domains of attraction of max stable laws under linear normalization. These results lead to the study of the max domains of attraction of distributions of products of independent random variables belonging to the max domain of attraction of max stable laws under power normalization.

On the equivalence of limit distributions of a sum and of a maximum sum of independent random variables

We prove the equivalence of the limit distributions of an appropriately centered and normalized sum and the maximum sum of independent random variables which have finite expectations. The result is an extension of a result of Kruglov (1999).

Bounding Probability of Small Deviation: A Fourth Moment Approach

In this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount (relative to the expected value), by means of the second and the fourth (central) moments of the random variable. In this particular context, many classical inequalities yield only trivial bounds. We obtain tight upper bounds by studying the primal-dual moments-generating conic optimization problems. As an application, we demonstrate that given the new probability bounds, a substantial sharpening and simplification of a recent result and its analysis by Feige (see Feige, U. 2006. On sums of independent random variables with unbounded variances, and estimating the average degree in a graph. SIAM J. Comput. 35 964–984) can be obtained; also, these bounds lead to new properties of the distribution of the cut values for the max-cut problem. We expect the new probability bounds to be useful in many other applications.

Integro-Local and Local Theorems on Normal and Large Deviations of the Sums of Nonidentically Distributed Random Variables in the Triangular Array Scheme

Gnedenko's local theorem and Stone–Shepp's integro-local theorem (see [L. A. Shepp, Ann. Math. Statist., 35 (1964), pp. 419–423], [C. Stone, Ann. Math. Statist., 36 (1965), pp. 546–551], [B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954]) for sums on independent identically distributed random variables are extended to the case when the summands are nonidentically distributed in a triangular array scheme. Under Cramer's condition on the random variables, we also obtain integro-local and local theorems that are valid in both normal and large deviations zones.

Asymptotic expansions of probability distributions

Asymtotic expansionsfor the probabilitydistributionsof sums of independentrandom variables are studied in this paper.

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.