Sums Of Independent Random Variables Research Articles

On the linear combination of independent logistic random variables

In this work the exact distribution of the linear combination of p independent logistic random variables is studied. It is shown that the exact distribution may be represented as a shifted infinite sum of independent random variables distributed as the difference of two independent Generalized Integer Gamma distributions. In addition, two near-exact approximations are developed for this distribution. Numerical studies are conducted to access the degree of precision and also the computational performance of these approximations. The developed methodology is used to derive near-exact approximations for the linear combination of independent generalized logistic random variables.

Variance of the sum of independent random variables in spheres

The sum of random variables (errors) is the key element both for its statistical study and for the estimation and control of errors in many scientific and technical applications. In this paper we analyze the sum of independent random variables (independent errors) in spheres. This type of errors are very important, for example, in quantum computing. We prove that, given two independent isotropic random variables in an sphere, X1 and X2, the variance verifies V(X1+X2)=V(X1)+V(X2)−V(X1)V(X2)2 and we conjecture that this formula is also true for non-isotropic random variables.

Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables

Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables

Characteristic Functions and Compactness of Distributions of Sums of Independent Random Variables

The sequences of distributions of centered sums of independent random variables are considered within the framework of the series scheme, without assuming the classical conditions for uniform asymptotic smallness and uniform limit constancy. Necessary and sufficient conditions are obtained for relative and stochastic compactness of such sequences in terms of the characteristic functions of summable random variables and with using their τ-centers.

An Estimate of the Absolute Constant in the Inequality for the Uniform Distance Between the Distributions of Sequential Sums of Independent Random Variables

Some of known inequalities for the uniform distance between distributions of sequential sums of independent identically distributed random variables are considered. In the case where the distribution F has 0 as the q-quantile, an upper bound for the absolute constant in the inequality is given.

Arak’s Inequalities for the Generalized Arithmetic Progressions

In 1980s, Arak has obtained powerful inequalities for the concentration functions of sums of independent random variables. Using these results, he has solved an old problem stated by Kolmogorov. In this paper, one of Arak’s results is modified to include generalized arithmetic progressions in the statement.

To the history of Saint-Petersburg school of Probability and Statistics. I. Limit theorems for sums of independent random variables

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as important relevant problems formulated in the second half of the twentieth century. The latter include the approximation of the distributions of sums of independent variables by infinitely divisible distributions, estimation of the accuracy of strong Gaussian approximation of such sums, and the limit theorems on the weak almost sure convergence of empirical measures generated by sequences of sums of independent random variables and vectors.

Criteria of Relative and Stochastic Compactness for Distributions of Sums of Independent Random Variables

We consider sequences of distributions of centered sums of independent random variables within the scheme of series without imposing the classical conditions of uniform asymptotic negligibility and...

Small Deviation Probabilities for Weighted Sum of Independent Random Variables with a Common Distribution Having a Power Decrease at Zero, under Minimal Moment Assumptions

We represent a certain weight condition such that the exact formula for the asymptotic behavior of small deviation probabilities for weighted sum of independent random variables with a common distribution, which has a power decay at zero, is valid under optimal moment assumptions.

Criteria of relative and stochastic compactness for distributions of sums of independent random variables

Рассматриваются последовательности распределений центрированных сумм независимых случайных величин в рамках схемы серий без классических условий равномерной предельной малости и равномерного предельного постоянства. Критерий относительной компактности таких последовательностей распределений был найден Г. Зигелем в работе [11] (1981 г.). В настоящей статье этот критерий сформулирован в более полной форме, а также приведено его новое доказательство, основанное на характеристических функциях. Также получен критерий более сильного свойства, введенного В. Феллером в [1], - стохастической компактности. Кроме того, приведено несколько новых формулировок критериев относительной и стохастической компактности указанных последовательностей распределений сумм в терминах характеристических функций суммируемых случайных величин.

On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables

Let $\{B_k\}_{k=1}^\infty , \{X_k\}_{k=1}^\infty $ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty $ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text {dist}} {=}\text{Ber} (p_k)$, with $\sum _{k=1}^\infty p_k=\infty $, and assume that $\{X_k\}_{k=1}^\infty $ satisfy $X_k>0$ and $\mu _k\equiv EX_k<\infty $. Let $M_n=\sum _{k=1}^np_k\mu _k$, assume that $M_n\to \infty $ and define the normalized sum of independent random variables $W_n=\frac 1{M_n}\sum _{k=1}^nB_kX_k$. We give a general condition under which $W_n\stackrel{\text {dist}} {\to }c$, for some $c\in [0,1]$, and a general condition under which $W_n$ converges weakly to a distribution from a family of distributions that includes the generalized Dickman distributions GD$(\theta ),\theta >0$. In particular, we obtain the following result, which reveals a strange domain of attraction to generalized Dickman distributions. Assume that $\lim _{k\to \infty }\frac{X_k} {\mu _k}\stackrel{\text {dist}} {=}1$. Let $J_\mu ,J_p$ be nonnegative integers, let $c_\mu ,c_p>0$ and let $ \mu _n\sim c_\mu n^{a_0}\prod _{j=1}^{J_\mu }(\log ^{(j)}n)^{a_j},\ p_n\sim c_p\big ({n^{b_0}\prod _{j=1}^{J_p}(\log ^{(j)}n)^{b_j}}\big )^{-1}, \ b_{J_p}\neq 0, $ where $\log ^{(j)}$ denotes the $j$th iterate of the logarithm. If \[ i.\ J_p\le J_\mu ;\\ ii.\ b_j=1, \ 0\le j\le J_p;\\ iii.\ a_j=0, \ 0\le j\le J_p-1,\ \text{and} \ \ a_{J_p}>0, \] then $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\frac 1{\theta }\text{GD} (\theta ),\ \text{where} \ \theta =\frac{c_p} {a_{J_p}}. $ Otherwise, $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\delta _c$, where $c\in \{0,1\}$ depends on the above parameters. We also give an application to the statistics of the number of inversions in certain random shuffling schemes.

Bounds for the Concentration Functions of Random Sums under Relaxed Moment Conditions

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The results obtained are extended to Poisson-binomial, binomial, and Poisson random sums. Under the same assumptions, bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are put forward for the accuracy of approximation of the concentration functions of geometric, negative binomial, and Sichel random sums by exponential, folded variance gamma, and folded Student distributions. Numerical estimates of all the constants involved are written down explicitly.

Group Testing Schemes From Codes and Designs

In group testing, simple binary-output tests are designed to identify a small number $t$ of defective items that are present in a large population of $N$ items. Each test takes as input a group of items and produces a binary output indicating whether the group is free of the defective items or contains one or more of them. In this paper, we study a relaxation of the combinatorial group testing problem. A matrix is called $(t,\epsilon )$ -disjunct if it gives rise to a nonadaptive group testing scheme with the property of identifying a uniformly random $t$ -set of defective subjects out of a population of size $N$ with false positive probability of an item at most $\epsilon $ . We establish a new connection between $(t,\epsilon )$ -disjunct matrices and error correcting codes based on the dual distance of the codes and derive estimates of the parameters of codes that give rise to such schemes. Our methods rely on the moments of the distance distribution of codes and inequalities for moments of sums of independent random variables. We also provide a new connection between group testing schemes and combinatorial designs.

An improved Bennett's inequality

ABSTRACTThis paper gives an improvement to Bennett's inequality for tail probability of sum of independent random variables, without imposing any additional condition. The improved version has a closed form expression. Using a refined arithmetic-geometric mean inequality, we further improve the obtained inequality. Numerical comparisons show that the proposed inequalities often improve the upper bound significantly in the far tail area, and these improvements get more prominent for larger sample size.

A Tight Upper Bound on the Second-Order Coding Rate of the Parallel Gaussian Channel With Feedback

This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: a peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases. The main contribution of this paper is a self-contained proof of an upper bound on the first- and second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss’ theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first- and second-order asymptotics.

A refined Hoeffding’s upper tail probability bound for sum of independent random variables

Using a refined arithmetic–geometric mean inequality, this paper gives an improved version of Hoeffding’s inequality which has a closed form and is easy to evaluate. Numerical simulation comparing the performance of the proposed inequality to the original version of Hoeffding’s inequality is also presented.

Products of random variables and the first digit phenomenon

We provide conditions on dependent and on non-stationary random variables Xn ensuring that the mantissa of the sequence of products ∏1nXk is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables.

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ3-metric measuring the difference between expectations of sufficiently smooth functions, like |·|3, of a sum of independent random variables X 1,..., X n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X 1,..., X n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).

Stochastic compactness of distributions of sums of independent random variables with finite variances∗

We consider sums of independent random variables within the scheme of series. We focus on the case where every summand has a zero mean and finite variance and sums have unit variances. We obtain a criterion of stochastic compactness (defined by W. Feller) for sequences of distributions of such sums. The condition of uniform asymptotic negligibility of summands is not supposed.

Information-Theoretic Lower Bounds for Distributed Function Computation

We derive information-theoretic converses (i.e., lower bounds) for the minimum time required by any algorithm for distributed function computation over a network of point-to-point channels with finite capacity, where each node of the network initially has a random observation and aims to compute a common function of all observations to a given accuracy with a given confidence by exchanging messages with its neighbors. We obtain the lower bounds on computation time by examining the conditional mutual information between the actual function value and its estimate at an arbitrary node, given the observations in an arbitrary subset of nodes containing that node. The main contributions include the following. First, a lower bound on the conditional mutual information via so-called small ball probabilities, which captures the dependence of the computation time on the joint distribution of the observations at the nodes, the structure of the function, and the accuracy requirement. For linear functions, the small ball probability can be expressed by Levy concentration functions of sums of independent random variables, for which tight estimates are available that lead to strict improvements over existing lower bounds on computation time. Second, an upper bound on the conditional mutual information via strong data processing inequalities, which complements and strengthens existing cutset-capacity upper bounds. Finally, a multi-cutset analysis that quantifies the loss (dissipation) of the information needed for computation as it flows across a succession of cutsets in the network. This analysis is based on reducing a general network to a line network with bidirectional links and self-links, and the results highlight the dependence of the computation time on the diameter of the network, a fundamental parameter that is missing from most of the existing lower bounds on computation time.