Some Optimal Conditions for the ASCLT.

Let X1,X2,… be independent random variables with EXk=0 and σk2:=EXk2<∞ (k≥1). Set Sk=X1+⋯+Xk and assume that sk2:=ESk2→∞. We prove that under the Kolmogorov condition |Xn|≤Ln,Ln=o(sn/(loglogsn)1/2)we have 1logsn2∑k=1nσk+12sk2fSksk→12π∫Rf(x)e-x2/2dxa.s.for any almost everywhere continuous function f:R→R satisfying |f(x)|≤eγx2, γ<1/2. We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process {Sn,n≥1} by a Wiener process.

Khinchin Families, Set Constructions, Partitions and Exponentials.

In this paper, we give a simple criterion to verify that functions of the form eg are in the Hayman class when g is a power series with nonnegative coefficients. Thus, using the Hayman and Báez-Duarte formulas, we obtain asymptotics for the coefficients of generating functions that arise in many examples of set construction in analytic combinatorics. This new criterion greatly simplifies the one obtained previously by the authors.

A central limit theorem for integer partitions into small powers.

The study of the well-known partition function p(n) counting the number of solutions to n=a1+⋯+aℓ with integers 1≤a1≤⋯≤aℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=a1α+⋯+aℓαwith 1≤a1<⋯

The Emergence of the Normal Distribution in Deterministic Chaotic Maps.

The central limit theorem states that, in the limits of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to attain a stable distribution. The condition of independence, however, only holds in real systems as an approximation. To extend the theorem to more general situations, previous studies have derived a version of the central limit theorem that also holds for variables that are not independent. Here, we present numerical results that characterize how convergence is attained when the variables being summed are deterministically related to one another through the recurrent application of an ergodic mapping. In all the explored cases, the convergence to the limit distribution is slower than for random sampling. Yet, the speed at which convergence is attained varies substantially from system to system, and these variations imply differences in the way information about the deterministic nature of the dynamics is progressively lost as the number of summands increases. Some of the identified factors in shaping the convergence process are the strength of mixing induced by the mapping and the shape of the marginal distribution of each variable, most particularly, the presence of divergences or fat tails.

Quantum entropy and central limit theorem.

We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a "maximal entropy principle in DV systems." We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a "second law of thermodynamics for quantum convolutions." We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the "magic gap," which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier.

A Note on Cumulant Technique in Random Matrix Theory.

We discuss the cumulant approach to spectral properties of large random matrices. In particular, we study in detail the joint cumulants of high traces of large unitary random matrices and prove Gaussian fluctuation for pair-counting statistics with non-smooth test functions.

Separation of mass spectra of hydrogen-deuterium exchanged ions obtained by electrospray of solutions of biopolymers with unknown primary structure.

The work is dedicated to further development of our described method for analyzing mass spectra of biomolecules acquired as a result of hydrogen-deuterium exchange reactions (HDXs). The modified method consists of separating HDX distributions via their approximations by a minimum number of components corresponding to independent H/D substitutions and independent charge carrier retentions in different spatial isoforms or conformations of biomolecules with unknown primary structures. In this case, neither the natural isotopic distribution nor the exact number of active sites involved in HDXs and H+ or D+ attachments can be determined in advance. Original H/D electrospray mass spectra of an apamin solution were taken from our previous work. In that work, taking into account the natural isotopic distribution of apamin molecules, three main conformations of apamin ions were found as a result of separating the H/D mass spectra of the apamin solution for the gas flow with the addition of about 10% ND3 molecules. Using the proposed modified method that does not require knowledge of the primary structure of the biomolecules gave similar results with slight deviations of calculated HDX distributions of the apamin ions from those obtained earlier. The maximum difference between mean values of the calculated HDX distributions for ions of the same charge in both cases does not exceed a few percent. In addition, HDX mass spectra of the apamin complex with an adduct of unknown structure were processed. Such analysis gave also three main fractions of ions with relatively large contributions when ND3 was injected into a radio-frequency quadrupole. In the absence of ND3 flow, the results of calculations for apamin and its complex were close to each other too. The formation of the apamin complex most probably in solution was confirmed by performed calculations.

Accuracy and Precision Improvement of Temperature Measurement Using Statistical Analysis/Central Limit Theorem.

This paper describes a method for increasing the accuracy and precision of temperature measurements of a liquid based on the central limit theorem. A thermometer immersed in a liquid exhibits a response with determined accuracy and precision. This measurement is integrated with an instrumentation and control system that imposes the behavioral conditions of the central limit theorem (CLT). The oversampling method exhibited an increasing measurement resolution. Through periodic sampling of large groups, an increase in the accuracy and formula of the increase in precision is developed. A measurement group sequencing algorithm and experimental system were developed to obtain the results of this system. Hundreds of thousands of experimental results are obtained and seem to demonstrate the proposed idea's validity.

Wave Function Realization of a Thermal Collision Model.

An efficient algorithm to simulate dynamics of open quantum system is presented. The method describes the dynamics by unraveling stochastic wave functions converging to a density operator description. The stochastic techniques are based on the quantum collision model. Modeling systems dynamics with wave functions and modeling the interaction with the environment with a collision sequence reduces the scale of the complexity significantly. The algorithm developed can be implemented on quantum computers. We introduce stochastic methods that exploit statistical characteristics of the model such as Markovianity, Brownian motion, and binary distribution. The central limit theorem is employed to study the convergence of distributions of stochastic dynamics of pure quantum states represented by wave vectors. By averaging a sample of functions in the distribution we prove and demonstrate the convergence of the dynamics to the mixed quantum state described by a density operator.

Robust Inference for Partially Observed Functional Response Data.

Irregular functional data in which densely sampled curves are observed over different ranges pose a challenge for modeling and inference, and sensitivity to outlier curves is a concern in applications. Motivated by applications in quantitative ultrasound signal analysis, this paper investigates a class of robust M-estimators for partially observed functional data including functional location and quantile estimators. Consistency of the estimators is established under general conditions on the partial observation process. Under smoothness conditions on the class of M-estimators, asymptotic Gaussian process approximations are established and used for large sample inference. The large sample approximations justify a bootstrap approximation for robust inferences about the functional response process. The performance is demonstrated in simulations and in the analysis of irregular functional data from quantitative ultrasound analysis.

The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications.

In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension d = 1 , 2 . Under mild assumptions, we provide L p -estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned L p -estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The L p -estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].

A central limit theorem concerning uncertainty in estimates of individual admixture.

The concept of individual admixture (IA) assumes that the genome of individuals is composed of alleles inherited from K ancestral populations. Each copy of each allele has the same chance qk to originate from population k, and together with the allele frequencies p in all populations at all M markers, comprises the admixture model. Here, we assume a supervised scheme, i.e. allele frequencies p are given through a reference database of size N, and q is estimated via maximum likelihood for a single sample. We study laws of large numbers and central limit theorems describing effects of finiteness of both, M and N, on the estimate of q. We recall results for the effect of finite M, and provide a central limit theorem for the effect of finite N, introduce a new way to express the uncertainty in estimates in standard barplots, give simulation results, and discuss applications in forensic genetics.

An epidemic model with short-lived mixing groups.

Almost all epidemic models make the assumption that infection is driven by the interaction between pairs of individuals, one of whom is infectious and the other of whom is susceptible. However, in society individuals mix in groups of varying sizes, at varying times, allowing one or more infectives to be in close contact with one or more susceptible individuals at a given point in time. In this paper we study the effect of mixing groups beyond pairs on the transmission of an infectious disease in an SIR (susceptible [Formula: see text] infective [Formula: see text] recovered) model, both through a branching process approximation for the initial stages of an epidemic with few initial infectives and a functional central limit theorem for the trajectories of the numbers of infectives and susceptibles over time for epidemics with many initial infectives. We also derive central limit theorems for the final size of (i) an epidemic with many initial infectives and (ii) a major outbreak triggered by few initial infectives. We show that, for a given basic reproduction number [Formula: see text], the distribution of the size of mixing groups has a significant impact on the probability and final size of a major epidemic outbreak. Moreover, the standard pair-based homogeneously mixing epidemic model is shown to represent the worst case scenario, with both the highest probability and the largest final size of a major epidemic.

Estimating Drift Parameters in a Sub-Fractional Vasicek-Type Process.

This study deals with drift parameters estimation problems in the sub-fractional Vasicek process given by dxt=θ(µ−xt)dt+dStH, with θ>0, µ∈R being unknown and t≥0; here, SH represents a sub-fractional Brownian motion (sfBm). We introduce new estimators θ^ for θ and µ^ for µ based on discrete time observations and use techniques from Nordin−Peccati analysis. For the proposed estimators θ^ and µ^, strong consistency and the asymptotic normality were established by employing the properties of SH. Moreover, we provide numerical simulations for sfBm and related Vasicek-type process with different values of the Hurst index H.

Functional delta residuals and applications to simultaneous confidence bands of moment based statistics.

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen's d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.

Approximating the Operating Characteristics of Bayesian Uncertainty Directed Trial Designs.

Bayesian response adaptive clinical trials are currently evaluating experimental therapies for several diseases. Adaptive decisions, such as pre-planned variations of the randomization probabilities, attempt to accelerate the development of new treatments. The design of response adaptive trials, in most cases, requires time consuming simulation studies to describe operating characteristics, such as type I/II error rates, across plausible scenarios. We investigate large sample approximations of pivotal operating characteristics in Bayesian Uncertainty directed trial Designs (BUDs). A BUD trial utilizes an explicit metric u to quantify the information accrued during the study on parameters of interest, for example the treatment effects. The randomization probabilities vary during time to minimize the uncertainty summary u at completion of the study. We provide an asymptotic analysis (i) of the allocation of patients to treatment arms and (ii) of the randomization probabilities. For BUDs with outcome distributions belonging to the natural exponential family with quadratic variance function, we illustrate the asymptotic normality of the number of patients assigned to each arm and of the randomization probabilities. We use these results to approximate relevant operating characteristics such as the power of the BUD. We evaluate the accuracy of the approximations through simulations under several scenarios for binary, time-to-event and continuous outcome models.

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application.

Confidence intervals for policy evaluation in adaptive experiments.

Adaptive experimental designs can dramatically improve efficiency in randomized trials. But with adaptively collected data, common estimators based on sample means and inverse propensity-weighted means can be biased or heavy-tailed. This poses statistical challenges, in particular when the experimenter would like to test hypotheses about parameters that were not targeted by the data-collection mechanism. In this paper, we present a class of test statistics that can handle these challenges. Our approach is to adaptively reweight the terms of an augmented inverse propensity-weighting estimator to control the contribution of each term to the estimator's variance. This scheme reduces overall variance and yields an asymptotically normal test statistic. We validate the accuracy of the resulting estimates and their CIs in numerical experiments and show that our methods compare favorably to existing alternatives in terms of mean squared error, coverage, and CI size.

Central Limit Theorem-Based Analysis Method for MicroRNA Detection with Solid-State Nanopores.

Although nanopore as a single-molecule sensing platform has proven its potential in various applications, data analysis of nanopores remains challenging. Herein, we introduce a method with increased accuracy in nanopore analysis based on the central limit theorem (CLT). An optimal voltage used in detection is determined from the standard deviations of blockage currents and time constants at various voltage biases. Compared with the conventional data analysis method, blockage signals processed with the CLT result in more concentrated distributions of blockage currents and durations. It allows fitting a Gaussian to the duration histogram and avoids the influence of bin sizes on time constants in duration analysis. The proposed method is further validated by applying it to detect isolated microRNAs with solid-state nanopores. Under the optimal voltage, different nucleic acids present in the isolation process are translocated through the nanopore. By processing the event signals with the CLT, all the nucleic acids including the microRNA are well differentiated. The method proposed here should also be applicable for sensing other biomolecules with the solid-state nanopores.