Sums Of Nonnegative Random Variables Research Articles

On the Strong Law of Large Numbers for Nonnegative Random Variables. With an Application in Survey Sampling

Strong laws of large numbers with arbitrary norming sequences for nonnegative not necessarily independent random variables are obtained. From these results we establish, among other things, stability results for weighted sums of nonnegative random variables. A survey sampling application is provided on strong consistency of the Horvitz--Thompson estimator and the ratio estimator.

Asymptotically Exact Approximations to Generalized Fading Sum Statistics

We propose a unified approach to approximate the probability density function and the cumulative distribution function of a sum of independent channel envelopes following generalized, and possibly mixed, fading models. In the proposed approach, the approximate sum distribution can be chosen out of a broad class of statistical models. More fundamentally, given any chosen model, the parameters of the approximate sum distribution are calibrated by matching its asymptotic behavior around zero to that of the exact sum. In a subsidiary fashion, one or more moments of the exact and approximate sums are also matched to one another if the approximate distribution has three or more parameters to be adjusted, respectively. As illustrated through many numerical examples, our approach outperforms existing ones that are solely based on moment matching, by yielding statistical approximations that are remarkably accurate at medium to high signal-to-noise ratio — a paramount operational regime for communications systems. Our results find applicability in several wireless applications where fading sums arise, and can be readily extended to accommodate sums of fading (power) gains and, in a broader context, generic sums of non-negative random variables.

Severity of risks threatening a class of systems

Stochastic models are generally recognised as extremely powerful analytical tools for investigating fundamental concepts and operations of the risk management process. The present paper aims to consider stochastic modelling efforts related to effective reduction of the severity of risks causing multiple damages to a class of systems. A new stochastic model is formulated which is shown to be a classical problem of risk measurement operations. The investigation of such a model is based on classical methods of probability distributions theory. A stochastic model having the form of a random sum of non-negative random variables is formulated. Sufficient conditions for evaluating the characteristic function of the proposed random sum are established. Moreover, applications of this random sum in evaluating the severity of a risk threatening a class of systems are provided. The difficulty of establishing properties which extend the practical applicability of the proposed random sum still remains. The formulated stochastic model provides a new original way to measure catastrophic risks and overcome the existing classical risk measurement stochastic models.

Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables

In this paper, asymptotic lower bounds of precise large deviations for non-random sums and random sums of nonnegative random variables (r.v.s) are derived under some fairly weak conditions. The obtained results are used to derive asymptotic lower bounds of precise large deviations in a multi-risk model. All the results we establish extend and improve the related existing results substantially.

Some Bounds on the Deviation Probability for Sums of Nonnegative Random Variables Using Upper Polynomials, Moment and Probability Generating Functions

We present several new bounds for certain sums of deviation probabilities involving sums of nonnegative random variables. These are based upon upper bounds for the moment generating functions of the sums. We compare these new bounds to those of Maurer [2], Bernstein [4], Pinelis [16], and Bentkus [3]. We also briefly discuss the infinitely divisible distributions case.

Karl Pearson’s meta-analysis revisited

This paper revisits a meta-analysis method proposed by Pearson [Biometrika 26 (1934) 425–442] and first used by David [Biometrika 26 (1934) 1–11]. It was thought to be inadmissible for over fifty years, dating back to a paper of Birnbaum [J. Amer. Statist. Assoc. 49 (1954) 559–574]. It turns out that the method Birnbaum analyzed is not the one that Pearson proposed. We show that Pearson’s proposal is admissible. Because it is admissible, it has better power than the standard test of Fisher [Statistical Methods for Research Workers (1932) Oliver and Boyd] at some alternatives, and worse power at others. Pearson’s method has the advantage when all or most of the nonzero parameters share the same sign. Pearson’s test has proved useful in a genomic setting, screening for age-related genes. This paper also presents an FFT-based method for getting hard upper and lower bounds on the CDF of a sum of nonnegative random variables.

On stability of sums of nonnegative random variables

We present new sufficient conditions for stability of sums of nonnegative random variables having finite moments of second order. We demonstrate that these conditions are nonimprovable in some sense. Bibliography: 4 titles.

Sum of Squared Shadowed-Rice Random Variables and its Application to Communication Systems Performance Prediction

The distribution of the sum of non-negative random variables plays an essential role in the performance analysis of diversity schemes for wireless communications over fading channels. While for common fading models such as the Rayleigh, Rice, and Nakagami, the performance of diversity systems is well understood, a minor attention has been devoted to the shadowed-Rice (SR) case, namely a Rice fading channel with fluctuating (e.g. random) Line of Sight (LOS) component. Indeed, the analytical performance evaluation of diversity systems on SR fading channels requires the availability of handy expressions for the distribution of the combined received power. To this end, the rationale of this paper is twofold: first, to evaluate the distribution of the sum of SR random variables, both for the case of independent as well as correlated LOS components, and then to carry out an extensive performance analysis of maximal ratio combining (MRC) detection scheme on SR fading channels.

On the Growth of Sums of Nonnegative Random Variables

Almost sure estimates of the growth of sums of nonnegative random variables are established. A generalization of author's result on the growth of sums of the indicators of arbitrary events is obtained. Bibliography: 10 titles.

A stochastic model for the cost of maintaining a risk frequency reduction operation

The paper is devoted to the formulation of a stochastic multiplicative model by making use of a nonnegative random variable and a geometric random sum of nonnegative random variables. Application of the model in evaluating the cost of maintaining a risk frequency reduction operation is provided. The paper also investigates the distribution of the model. More precisely, sufficient conditions are established for embedding the distribution of the model into a very important class of infinitely divisible and unimodal scale mixtures of distributions.

Complementary distribution function for the sum of non-negative random variables: A reliability application

Abstract Many organization problems arising in reliability and risk assessment analysis (and in general in rare event probability computations) require challenging calculations of very small (10−2 to 10−6) probability values. This paper considers the distribution function D 1,N (x) of the sum of N non-negative continuous random variables and the relevant complementary distribution function, C 1,N =1−D 1,N . The numerical examples refer to the case of N independent random variables but the whole analysis is presented without this restriction. The method developed proves capable of evaluating very small probabilities with the accuracy appropriate to applications and very short computing times. Comparison with analytical results are provided for simple distributions, and an application to human reliability is demonstrated.

On divergence and convergence of sums of nonnegative random variables

Abstract If X1, X2,… are exponentially distributed random variables thenσ∞k= 1 Xk=∞ with probability 1 iff σ∞k= 1 EXk=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI‐MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.