Non-identical Random Variables Research Articles

Non-asymptotic bounds for percentiles of independent non-identical random variables

This note displays an interesting phenomenon for the percentiles of independent but non-identical random variables. Let X1,…,Xn be independent random variables obeying non-identical continuous distributions and X(1)≥⋯≥X(n) be the corresponding order statistics. For p∈(0,1), we investigate the 100(1−p)%th percentile X(⌊pn⌋) and prove the non-asymptotic bounds for X(⌊pn⌋). In particular, for a wide class of distributions, we discover an intriguing connection between their median and the harmonic mean of the associated standard deviations. For example, if Xk∼N(0,σk2) for k=1,…,n and p=12, we show that its median |Med(X1,…,Xn)|=OP(n1∕2⋅(∑k=1nσk−1)−1) as long as {σk}k=1n satisfy certain mild non-dispersion property.

Phase transitions in the one-dimensional Coulomb gas ensembles

We consider the system of particles on a finite interval with pairwise nearest neighbours interaction and external force. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31–36] to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature. The proofs rely on a conditional central limit theorem for nonidentical random variables, which has an interest on its own.

Super Generalized Central Limit Theorem —Limit Distributions for Sums of Non-identical Random Variables with Power Laws—

In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the mathematical characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical stochastic processes with power-laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws such that the sums of the logarithmic return of non-identical stock price fluctuations follow stable distributions.

Diversity in cascaded N*Nakagami channels

Short term fading conditions that are worse than those existing in typical Nakagami channels can be modeled using the cascaded approach where the received signal-to-noise ratio is treated as the product of N identical or nonidentical random variables that are either dependent or independent. Cascaded channels also permit the modeling of fading seen in amplify and forward relaying channels as well as keyhole channels. The simplest of these models, referred to as the N*Nakagami model, assumes that the received signal to noise ratio is the product of N independent and identically distributed gamma variables. The fading mitigation achieved in such cascaded channels through diversity is studied in this work. Through analytical approach and simulation, the performance of wireless channels is examined for the selection combining, equal gain combining, generalized selection combining, and maximal ratio combining algorithms. Results demonstrate the existence of severe degradations in cascaded fading channels and the comparative usefulness of the various algorithms in improving the data transmission capabilities. Even though the analysis was undertaken for the case of a coherent binary shift keying modem, results can easily be extended to other modems.

On properties of progressively Type-II censored order statistics arising from dependent and non-identical random variables

In this paper, we study progressively Type-II censored order statistics arising from identical as well as non-identical units under test which are jointly distributed according to an Archimedean copula with completely monotone generator (PCOSDNARCM-II). Density, distribution and joint density functions of PCOSDNARCM-II are all derived. For certain special cases, more explicit expressions are presented. Some interesting recurrence relations and transformational properties are also established. Results established here contain the results by Balakrishnan and Cramer [5] as particular cases. Finally, some examples of PCOSDNARCM-II are also provided.

A Laplace transform method for order statistics from nonidentical random variables and its application in Phase-type distribution

In this paper, we derive a method for obtaining the Laplace transform of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). A survey of the most important properties, applications and the o.s. of a Phase-type (PH) distribution are also presented. Two illustrative examples are provided.

Large deviations and moderate deviations for sums of negatively dependent random variables

In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (−∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.

Computing the moments of order statistics from nonidentical random variables

In this paper, we establish new representations, identities and recurrence relations of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). These recurrence relations will enable one to compute all moments of all o.s. in a simple manner. Applications for some known distributions are given.