Family Of Random Variables Research Articles

Taylor’s Law for Some Infinitely Divisible Probability Distributions from Population Models

In a family of random variables, Taylor’s law or Taylor’s power law of fluctuation scaling is a variance function that gives the variance $$\sigma ^{2}>0$$ of a random variable (rv) X with expectation $$\mu >0$$ as a power of $$\mu $$ : $$\sigma ^{2}=A\mu ^{b}$$ for finite real $$A>0,\ b$$ that are the same for all rvs in the family. Equivalently, TL holds when $$\log \sigma ^{2}=a+b\log \mu ,\ a=\log A$$ , for all rvs in some set. Here we analyze the possible values of the TL exponent b in five families of infinitely divisible two-parameter distributions and show how the values of b depend on the parameters of these distributions. The five families are Tweedie–Bar-Lev–Enis, negative binomial, compound Poisson-geometric, compound geometric-Poisson (or Pólya-Aeppli), and gamma distributions. These families arise frequently in empirical data and population models, and they are limit laws of Markov processes that we exhibit in each case.

On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

We study the local eigenvalue statistics $\xi_{\omega,E}^N$ associated with the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as $N^\alpha$, for $0 < \alpha < \frac{1}{2}$. We consider the limit points associated with the random variables $\xi_{\omega,E}^N [I]$, for $I \subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random variables with $0 \leq \alpha < \frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every $E \in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables $\xi_{\omega,E}^N [I]$ and associated quantities related to the intensities, as $N$ tends towards infinity, and employs known localization bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for $0 < \alpha < \frac{1}{2}$, we prove that any nontrivial limit points of the random variables $\xi_{\omega,E}^N [I]$ are distributed according to Poisson distributions.

Geometrical smeariness – a new phenomenon of Fréchet means

In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fréchet mean fluctuates around the population mean asymptotically at a scale n−τ with exponent τ<1/2 with a non-normal distribution if the probability density at the antipodal point of the mean is 12π. The author and his collaborator recently discovered that τ=1/6 for some random variables on higher dimensional spheres. In this article we show that, even more surprisingly, the phenomenon on spheres of higher dimension is qualitatively different from that on the circle, as it depends purely on geometrical properties of the space, namely its curvature, and not on the density at the antipodal point. This gives rise to the new concept of geometrical smeariness. In consequence, the sphere can be deformed, say, by removing a neighborhood of the antipodal point of the mean and gluing a flat space there, with a smooth transition piece. This yields smeariness on a manifold, which is diffeomorphic to Euclidean space. We give an example family of random variables with 2-smeary mean, that is, with τ=1/6, whose range has a hole containing the cut locus of the mean. The hole size exhibits a curse of dimensionality as it can increase with dimension, converging to the whole hemisphere opposite a local Fréchet mean. We observe smeariness in simulated landmark shapes on Kendall pre-shape space and in real data of geomagnetic north pole positions on the two-dimensional sphere.

Asymptotic results for families of random variables having power series distributions

Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].

H∞ Proportional-Integral State Estimation for T-S Fuzzy Systems Over Randomly Delayed Redundant Channels With Partly Known Probabilities.

In this article, we consider the H∞ proportional-integral (PI) state estimation (SE) problem for discrete-time T-S fuzzy systems subject to transmission delays, external disturbances, and redundant channels. Multiple redundant communication channels are utilized between the sensors and the remote estimator to enhance the reliability of data transmissions. In order to characterize the transmission delays in network-based communication, a family of random variables with partly known probabilities, which are independent and identically distributed, is adopted to describe the random behavior of the transmission delays with the redundant channels. The objective of this work is to put forward a PI state estimator such that the dynamics of the estimation error is exponentially mean-square stable and satisfies the prescribed H∞ performance index of the disturbance attenuation/rejection. By employing the stochastic analysis approach, the error dynamics of the SE under the proposed state estimator is analyzed and sufficient conditions are obtained to ensure the existence of the required PI state estimator. Furthermore, the desired estimator parameters are derived by solving a nonlinear optimization problem. Finally, two simulation examples are exploited to demonstrate the validity of the proposed SE scheme.

Probabilistic Reactive Power Capability Charts at DSO/TSO Interface

Hosting an increasing share of distributed generation and flexible consumers, distribution systems are being more often considered as providers of ancillary services among which are reactive power support and voltage control. For the purpose of reliable and efficient planning and operation of a transmission system, information about reactive power capabilities of connected distribution systems would be beneficial. These capabilities are subject to uncertainties especially with a presence of intermittent distributed generation and stochastic loads. To capture these uncertainties, this paper introduces probabilistic capability charts using the concept of probabilistic curves. These curves analytically represent reactive power support limits of distribution systems as a family of random variables with their associated probabilistic density functions. As such, they carry more information about states of the system than deterministic curves. The proposed capability charts can be easily applied in probabilistic studies of the overlaying transmission system. By providing analytical expressions of stochastic power injections of connected distribution systems, the use of these charts can speed up time and memory demanding probabilistic calculations of transmission systems planning and operation studies. Not containing sensitive system information, the charts can be used to exchange data between distribution and transmission system operators.

Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

The combinatorics of the colliding bullets

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let (Vi)1 ≤ i ≤ n be an i.i.d. family of random variables with uniform distribution on [0,1]; shoot n bullets one after another at times 1,2,…,n, where the ith bullet has speed Vi. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalization of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.

The computation of the probability density and distribution functions for some families of random variables by means of the Wynn-ρ accelerated Post-Widder formula

We propose the suitable use of the Post-Widder inversion formula for Laplace transforms – coupled with the Wynn’s ρ-algorithm for accelerating sequences – in order to evaluate (up to the desired accuracy) the probability density function and the distribution function of a large collection of random variables. The method is illustrated on the Tweedie law and the tempered positive Linnik law. In addition, a further application to some laws arising in the context of Brownian motion is considered.

On the maximum of the CβE field

In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$\beta$E). More precisely, if $X_n$ is this characteristic polynomial and $\mathbb{U}$ the unit circle, we prove that: $$\sup_{z \in \mathbb{U} } \Re \log X_n(z) = \sqrt{\frac{2}{\beta}} \left(\log n - \frac{3}{4} \log \log n + \mathcal{O}(1) \right)\ ,$$ as well as an analogous statement for the imaginary part. The notation $\mathcal{O}(1)$ means that the corresponding family of random variables, indexed by $n$, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the case where $\beta$ equals to $2$, which corresponds to the CUE field.

On probability and logic

Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations by means of stochastic valuations. A stochastic valuation is a stochastic process, that is a family of random variables one for each propositional symbol. With stochastic valuations we are able to cope with a countably infinite set of propositional symbols. A notion of probabilistic entailment enjoying desirable properties of logical consequence is defined and shown to collapse into the classical entailment when the propositional language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the decidability of the theory of real closed ordered fields.

Uniformity of hitting times of the contact process

For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times t(x), defined for each site x as the first time at which it becomes infected. First, the family of random variables (t(x) - t(y))/vertical bar x-y vertical bar, indexed by x not equal y in Z(d) , is stochastically tight. Second, for each epsilon > 0 there exists x such that, for infinitely many integers n, t(nx) < t((n + 1)x) with probability larger than 1-epsilon. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchanc (2012).

Weighted dependency graphs

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables. To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vallee on the asymptotic normality of subword counts in random texts generated by a Markovian source.

Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

Distribution of the Maximum and Minimum of a Random Number of Bounded Random Variables

We study a new family of random variables that each arise as the distribution of the maximum or minimum of a random number N of i.i.d. random variables X1, X2,…, XN, each distributed as a variable X with support on [0, 1]. The general scheme is first outlined, and several special cases are studied in detail. Wherever appropriate, we find estimates of the parameter θ in the one-parameter family in question.

Improvement of the k-nn Entropy Estimator with Applications in Systems Biology

In this paper, we investigate efficient estimation of differential entropy for multivariate random variables. We propose bias correction for the nearest neighbor estimator, which yields more accurate results in higher dimensions. In order to demonstrate the accuracy of the improvement, we calculated the corrected estimator for several families of random variables. For multivariate distributions, we considered the case of independent marginals and the dependence structure between the marginal distributions described by Gaussian copula. The presented solution may be particularly useful for high dimensional data, like those analyzed in the systems biology field. To illustrate such an application, we exploit differential entropy to define the robustness of biochemical kinetic models.

Reliability estimation of structures under stochastic loading—A case study on nuclear piping

Generally structures are subjected to different types of loadings throughout their life time. These loads can be either discrete in nature or continuous in nature and also these can be either stationary or non stationary processes. This means that the structural reliability analysis not only considers random variables but also considers random variables which are functions of time, referred to as stochastic processes. A stochastic process can be viewed as a family of random variables. When a structure is subjected to a random loading, based on the stresses developed in the structure and failure criteria the failure probability can be estimated. In practice the structures are designed with higher factor of safety to take care of such random loads. In such cases the structure will fail only when the random loads are cyclic in nature. In traditional reliability analysis, the variation in the load is treated as a random variable and to account for the number of occurrences of the loading the concept of extreme value theory is used. But with this method one is neglecting the damage accumulation that will take place from one loading to another loading. Hence, in this paper, a new way of dealing with these types of problems has been discussed by using the concept of stochastic fatigue loading. The random loading has been considered as earthquake loading. The methodology has been demonstrated with a case study on nuclear power plant piping.

Stochastic order characterization of uniform integrability and tightness

We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. In particular, we show that, whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics.

Generalized Galois Numbers, Inversions, Lattice Paths, Ferrers Diagrams and Limit Theorems

Bliem and Kousidis recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables, using inversions in random words, random lattice paths and random Ferrers diagrams, and use these to give new proofs of limit theorems as well as some further limit results.

Optimal multiple stopping time problem

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $\phi$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$. This new reward $\phi$ is not a right-continuous adapted process as in the classical case, but a family of random variables. For such a reward, we prove a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).