Sums Of Independent Random Variables Research Articles

Local Limit Theorems for Smoothed Bernoulli and Other Convolutions

We explore the asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.

Exact weak laws of large numbers with applications to ratios of random variables

We study convergence in probability of weighted sums of independent random variables which are not necessarily identically distributed. The results obtained are applied to ratios of independent random variables and ratios of smallest order statistics.

STATISTICAL TESTING TECHNIQUE FOR COMPARISON MACHINE LEARNING MODELS PERFORMANCE

The statistical testing technique is considered to compare the metrics values of machine learning models on a test set. Since the values of metrics depend not only on the models, but also on the data, it may turn out that different models are the best on different test sets. For this reason, the traditional approach to comparing the values of metrics on a test set is often not enough. Sometimes a statistical comparison of the results obtained on the basis of cross-validation is used, but in this case it is impossible to guarantee the independence of the obtained measurements, which does not allow the use of the Student's t-test. There are criteria that do not require independent measurements, but they have less power. For additive metrics, a technique is proposed in this paper, when a test sample is divided into N parts, on each of which the values of the metrics are calculated. Since the value on each part is obtained as the sum of independent random variables, according to the central limit theorem, the obtained metrics values on each of the N parts are realizations of the normally distributed random variable. To estimate the required sample size, it is proposed to use normality tests and build quantile– quantile plots. You can then use a modification of the Student's t-test to conduct a statistical test comparing the mean values of the metrics. A simplified approach is also considered, in which confidence intervals are built for the base model. A model whose metric values do not fall into this interval works differently from the base model. This approach reduces the amount of computations needed, however, an experimental analysis of the binary cross-entropy metric for CTR (Click-Through Rate) prediction models showed that it is more rough than the first one.

Optimal Stopping Strategies in the Game “The Price Is Right”

The popular TV show “The Price Is Right” is an attractive source of modeling the strategic behavior in a competitive environment for a specific reward. In this study, the structure of the show is used as a basis for several game-theoretic settings. We consider a noncooperative optimal stopping game for a finite number of players. Each player earns points by observing the sums of independent random variables uniformly distributed on the unit interval. At each step, the player must decide whether to stop or continue the game. The winner is the player with the maximum score not exceeding unity. If the scores of all players exceed this limit, the winner is the player with the lowest score. We characterize the optimal strategies of the players in the multi-step version of the game with complete information about the scores of the previous players. We also compare the optimal strategies and payoffs of the players in the games with complete information and with no information. The notion of information price is introduced.

Moments of Random Variables: A Systems-Theoretic Interpretation

Moments of continuous random variables admitting a probability density function are studied. We show that, under certain assumptions, the moments of a random variable can be characterized in terms of a Sylvester equation and of the steady-state output response of a specific interconnected system. This allows to interpret well-known notions and results of probability theory and statistics in the language of systems theory, including the sum of independent random variables, the notion of mixture distribution and results from renewal theory. The theory developed is based on tools from center manifold theory, the theory of the steady-state response of nonlinear systems, and the theory of output regulation. Our formalism is illustrated by means of several examples and can be easily adapted to the case of discrete and multivariate random variables.

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.

An expansion for the number of partitions of an integer

We consider p(n) the number of partitions of a natural number n, starting from an expression derived by Baez-Duarte in (Adv Math 125(1):114–120, 1997) by relating its generating function f(t) with the characteristic functions of a family of sums of independent random variables indexed by t. The asymptotic formula for p(n) follows then from a local central limit theorem as $$t\uparrow 1$$ suitably with $$n\rightarrow \infty $$. We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as $$t\uparrow 1$$. Those include the generating function f and the cumulants of the random variables. After developing an asymptotic series expansion for the integral term, we obtain an expansion for p(n) that can be simplified as follows: for each $$N>0$$, $$\begin{aligned} p(n)=\frac{2\pi ^2}{3\sqrt{3}} \,\frac{\text{ e }^{r_n}}{(1+2\,r_n)^2}\, \left( 1-\sum _{\ell = 1}^N\,\frac{D_\ell }{(1+2\,r_n)^\ell }+\,\mathcal R_{N+1}\,\right) . \end{aligned}$$The coefficients $$D_\ell $$ are positive and have simple expressions as finite sums of combinatorial numbers, $$r_n=\sqrt{\frac{2\pi ^2}{3}\,(n-\frac{1}{24})+\frac{1}{4}}$$ and the remainder satisfies $$n^{N/2}\,{\mathcal {R}}_{N+1} \rightarrow 0$$ as $$n\rightarrow \infty $$. The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.

Small deviations of sums of independent random variables

A well-known discovery of Feige's is the following [3]: Let X1,…,Xn be nonnegative independent random variables, with E[Xi]≤1∀i, and let X=∑i=1nXi. Then for any n,Pr⁡[X<E[X]+1]≥α>0, for some α≥1/13. This bound was later improved to 1/8 by He, Zhang, and Zhang [5]. By a finer consideration of the first four moments, we further improve the bound to approximately .14. The conjectured true bound is 1/e≃.368.

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.

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).

M/M/1 Model With Unreliable Service and a Working Vacation

We derive an explicit closed form of the stationary distribution of an M/M/1 queue with unreliable service and a working vacation. We also show that the work in (Patterson &amp;amp; Korzeniowski, 2018) can be obtained as a special case of this model. Future work remains to be done; specifically, it may be possible to use the explicit stationary distribution given here to decompose the queue length into the sum of independent random variables. Consequently, it may then be possible to utilize Little&amp;rsquo;s Law (Little, 1961) to decompose the customer waiting time as well.

Series representations for densities functions of a family of distributions—Application to sums of independent random variables

Series representations for several density functions are obtained as mixtures of generalized gamma distributions with discrete mass probability weights, by using the exponential expansion and the binomial theorem. Based on these results, approximations based on mixtures of generalized gamma distributions are proposed to approximate the distribution of the sum of independent random variables, which may not be identically distributed. The applicability of the proposed approximations are illustrated for the sum of independent Rayleigh random variables, the sum of independent gamma random variables, and the sum of independent Weibull random variables. Numerical studies are presented to assess the precision of these approximations.

Classification by Sparse Neural Networks.

The choice of dictionaries of computational units suitable for efficient computation of binary classification tasks is investigated. To deal with exponentially growing sets of tasks with increasingly large domains, a probabilistic model is introduced. The relevance of tasks for a given application area is modeled by a product probability distribution on the set of all binary-valued functions. Approximate measures of network sparsity are studied in terms of variational norms tailored to dictionaries of computational units. Bounds on these norms are proven using the Chernoff-Hoeffding bound on sums of independent random variables that need not be identically distributed. Consequences of the probabilistic results for the choice of dictionaries of computational units are derived. It is shown that when a priori knowledge of a type of classification tasks is limited, then the sparsity may be achieved only at the expense of large sizes of dictionaries.

Approximation of the Evolution Operator by Expectations of Functionals of Sums of Independent Random Variables

A method of probabilistic approximation of the operator $e^{-itH}$, where $H = -\frac{1}{2}\,\frac{d^2}{dx^2}+V(x), V\in L_\infty({\mathbf{R}})$, in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of sums of independent identically distributed random variables.

Scaling limits of population and evolution processes in random environment

We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.

Approximation of the evolution operator by expectations of functionals of sums of independent random variables

Предложен способ вероятностной аппроксимации в смысле сильной операторной сходимости оператора $e^{-itH}$, где $H=-\frac{1}{2} \frac{d^2}{dx^2}+V(x)$, $V\in L_\infty(\mathbf R)$. Аппроксимирующие операторы имеют вид математических ожиданий функционалов от сумм независимых одинаково распределенных случайных величин.

Supermodular ordering of Poisson and binomial random vectors by tree-based correlations

We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.

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.