Real-valued Random Variables Research Articles

WHEN NO PRICE IS RIGHT

Abstract In this paper, we show how to represent a non-Archimedean preference over a set of random quantities by a nonstandard utility function. Non-Archimedean preferences arise when some random quantities have no fair price. Two common situations give rise to non-Archimedean preferences: random quantities whose values must be greater than every real number, and strict preferences between random quantities that are deemed closer in value than every positive real number. We also show how to extend a non-Archimedean preference to a larger set of random quantities. The random quantities that we consider include real-valued random variables, horse lotteries, and acts in the theory of Savage. In addition, we weaken the state-independent utility assumptions made by the existing theories and give conditions under which the utility that represents preference is the expected value of a state-dependent utility with respect to a probability over states.

Deviation inequalities for dependent sequences with applications to strong approximations

In this paper, we give precise rates of convergence in the strong invariance principle for stationary sequences of bounded real-valued random variables satisfying weak dependence conditions. One of the main ingredients is a new Fuk-Nagaev type inequality for a class of weakly dependent sequences. We describe also several classes of processes to which our results apply.

Regular variation in Hilbert spaces and principal component analysis for functional extremes

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm ‖X‖ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal ‘directions’ of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert–Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

Quantum properties of classical Pearson random variables

This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable [Formula: see text] with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of [Formula: see text] as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable [Formula: see text] and characterize type-[Formula: see text] and type-[Formula: see text] real-valued random variables. We prove that a necessary condition for [Formula: see text] to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type.

Using spatial ordinal patterns for non-parametric testing of spatial dependence

We analyze data occurring in a regular two-dimensional grid for spatial dependence based on spatial ordinal patterns (SOPs). After having derived the asymptotic distribution of the SOP frequencies under the null hypothesis of spatial independence, we use the concept of the type of SOPs to define the statistics to test for spatial dependence. The proposed tests are not only implemented for real-valued random variables, but a solution for discrete-valued spatial processes in the plane is provided as well. The performances of the spatial-dependence tests are comprehensively analyzed by simulations, considering various data-generating processes. The results show that SOP-based dependence tests have good size properties and constitute an important and valuable complement to the spatial autocorrelation function. To be more specific, SOP-based tests can detect spatial dependence in non-linear processes, and they are robust with respect to outliers and zero inflation. To illustrate their application in practice, two real-world data examples from agricultural sciences are analyzed.

Oscillating neural circuits: Phase, amplitude, and the complex normal distribution.

Multiple oscillating time series are typically analyzed in the frequency domain, where coherence is usually said to represent the magnitude of the correlation between two signals at a particular frequency. The correlation being referenced is complex-valued and is similar to the real-valued Pearson correlation in some ways but not others. We discuss the dependence among oscillating series in the context of the multivariate complex normal distribution, which plays a role for vectors of complex random variables analogous to the usual multivariate normal distribution for vectors of real-valued random variables. We emphasize special cases that are valuable for the neural data we are interested in and provide new variations on existing results. We then introduce a complex latent variable model for narrowly band-pass-filtered signals at some frequency, and show that the resulting maximum likelihood estimate produces a latent coherence that is equivalent to the magnitude of the complex canonical correlation at the given frequency. We also derive an equivalence between partial coherence and the magnitude of complex partial correlation, at a given frequency. Our theoretical framework leads to interpretable results for an interesting multivariate dataset from the Allen Institute for Brain Science.

The Cauchy Distribution in Information Theory.

The Gaussian law reigns supreme in the information theory of analog random variables. This paper showcases a number of information theoretic results which find elegant counterparts for Cauchy distributions. New concepts such as that of equivalent pairs of probability measures and the strength of real-valued random variables are introduced here and shown to be of particular relevance to Cauchy distributions.

The Hsu–Robbins–Erdös theorem for the maximum partial sums of quadruplewise independent random variables

Etemadi (1981) [10] and Rio (1995) [27] provided proofs of the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers under optimal moment conditions without using the Kolmogorov-type maximal inequalities. While this famous result holds for sequences of pairwise independent identically distributed real-valued random variables, a closely related result, the Hsu–Robbins–Erdös strong law of large numbers may fail if the underlying random variables are only assumed to be pairwise independent identically distributed. This note develops Rio's method and uses an approximation technique to establish the Hsu–Robbins–Erdös strong law of large numbers for the maximum partial sums of quadruplewise independent identically distributed random variables. We consider random variables taking values in a real separable Banach space X, but the main result is new even when X is the real line. Previous contributions so far considered the complete convergence of the partial sums or restricted to dependence structures satisfying a Kolmogorov-type maximal inequality.

A Look at the Primary Order Preserving Properties of Stochastic Orders: Theorems, Counterexamples and Applications in Cognitive Psychology

In this paper, we prove that for a set of ten univariate stochastic orders including the usual order, a univariate stochastic order preserves either both, one or none of additivity and multiplication properties over the vector space of real-valued random variables. Then, classifying participant’s quickness in a mental chronometry trial to “weakly faster” and “strongly faster”, we use the above results for the usual stochastic order to establish necessary and sufficient conditions for a participant to be strongly faster than the other in terms of the fitted Wald, Exponentially modified Wald(ExW), and Exponentially modified Gaussian(ExG) distributional parameters. This research field remains uncultivated for other univariate stochastic orders and in several directions.

Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy.

This paper studies the properties of the derivatives of differential entropy in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for , , while McKean conjectured a stronger statement, whereby . Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: , where n denotes that is a random vector taking values in , and similarly, . In this paper, we prove some new multivariate cases: . Motivated by our results, we further propose a weaker version of McKean’s conjecture , which is implied by and implies . We prove some multivariate cases of this conjecture under the log-concave condition: and . A systematic procedure to prove is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.

Asymptotic risk decomposition for regularly varying distributions with tail dependence

In this paper we investigate the limiting behaviour of Conditional Tail Expectation (CTE) and its decomposition for a sum of real-valued tail-dependent random variables with regularly varying distributions. Asymptotic proportions to the corresponding Value at Risk (VaR) measures are obtained for a flexible dependence structure. For a certain practical case considering an investment portfolio exact formulas are derived and sensitivity to model parameters is analysed. We also carry out a simulation study verifying our results and revealing the speed of convergence for different values of parameters.

Large-deviation results for triangular arrays of semiexponential random variables

AbstractAsymptotics deviation probabilities of the sum $S_n=X_1+\dots+X_n$ of independent and identically distributed real-valued random variables have been extensively investigated, in particular when $X_1$ is not exponentially integrable. For instance, Nagaev (1969a, 1969b) formulated exact asymptotics results for $\mathbb{P}(S_n>x_n)$ with $x_n\to \infty$ when $X_1$ has a semiexponential distribution. In the same setting, Brosset et al. (2020) derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large-deviation theory and making the rate function at the transition explicit. In this paper we exhibit the same asymptotic behavior for triangular arrays of semiexponentially distributed random variables.

The fundamental theorem of affine geometry in regular L0-modules

Let (Ω,F,P) be a probability space and L0(F) the algebra of equivalence classes of real-valued random variables defined on (Ω,F,P). A left module M over the algebra L0(F) (briefly, an L0(F)-module) is said to be regular if x=y for any given two elements x and y in M such that there exists a countable partition {An,n∈N} of Ω to F such that I˜An⋅x=I˜An⋅y for each n∈N, where IAn is the characteristic function of An and I˜An its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular L0(F)-modules: let V and V′ be two regular L0(F)-modules such that V contains a free L0(F)-submodule of rank 2, if T:V→V′ is stable and invertible and maps each L0-line segment to an L0-line segment, then T must be L0-affine.

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.

Novel Data Augmentation Employing Multivariate Gaussian Distribution for Neural Network-Based Blood Pressure Estimation

In this paper, we propose a novel data augmentation technique employing multivariate Gaussian distribution (DA-MGD) for neural network (NN)-based blood pressure (BP) estimation, which incorporates the relationship between the features in a multi-dimensional feature vector to describe the correlated real-valued random variables successfully. To verify the proposed algorithm against the conventional algorithm, we compare the results in terms of mean error (ME) with standard deviation and Pearson correlation using 110 subjects contributed to the database (DB) which includes the systolic BP (SBP), diastolic BP (DBP), photoplethysmography (PPG) signal, and electrocardiography (ECG) signal. For each subject, 3 times (or 6 times) measurements are accomplished in which the PPG and ECG signals are recorded for 20 s. And, to compare with the performance of the BP estimation (BPE) using the data augmentation algorithms, we train the BPE model using the two-stage system, called the stacked NN. Since the proposed algorithm can express properly the correlation between the features than the conventional algorithm, the errors turn out lower compared to the conventional algorithm, which shows the superiority of our approach.

A Semicircle Law for Derivatives of Random Polynomials

Abstract Let $x_1, \dots , x_n$ be $n$ independent and identically distributed real-valued random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots , x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty $, the $(n-\ell )-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval, a suitable rescaling of $p_n^{(n-\ell )}$ starts behaving like the $\ell -$th probabilists’ Hermite polynomial subject to a random shift. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

Real Zeros of Random Cosine Polynomials with Palindromic Blocks of Coefficients

It is well known that a random cosine polynomial \({V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} \), x ∈ (0, 2π), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has \(2n/\sqrt 3 \) expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree n) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length ℓ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is \({{\rm{K}}_\ell} \times 2n/\sqrt 3 \), where the constant Kℓ (depending only on ℓ) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.

On the Bernstein-von Mises theorem for the Dirichlet process

We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.

Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure

In this paper we consider the sum Snξ≔ξ1+…+ξn of (possibly dependent and nonidentically distributed) real-valued random variables ξ1,…,ξn with dominatedly varying distributions. Assuming that the ξk’s follow the dependence structure, similar to the asymptotic independence, we obtain the asymptotic lower and upper bounds for the tail moment E((Snξ)m1{Snξ>x}), where m is a nonnegative integer, improving the bounds of Leipus et al. (2019). We also consider the case of nonnegative random variables. Using the obtained results, we get the asymptotic estimations for the Haezendonck–Goovaerts risk measure in two examples of sums with regularly varying and dominatedly varying (but not regularly varying) increments.