Conditional Central Limit Theorem Research Articles

The intransitive dice kernel: 1x≥y-1x≤y4-3(x-y)(1+xy)8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{\\mathbbm {1}_{x\\ge y}-\\mathbbm {1}_{x\\le y}}{4} - \\frac{3(x-y)(1+xy)}{8}$$\\end{document}

Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability 1/4+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/4+o(1)$$\\end{document} and the probability a random pair of dice tie tends toward αn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha n^{-1}$$\\end{document} for an explicitly defined constant α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. This extends and sharpens the recent results of Polymath regarding the balanced sequence model. We further show the distribution of larger tournaments converges to a universal tournamenton in both models. This limit naturally arises from the discrete spectrum of a certain skew-symmetric operator (given by the kernel in the title acting on L2([-1,1])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2([-1,1])$$\\end{document}). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that Ai\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_i$$\\end{document} beats Ai+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{i+1}$$\\end{document} for 1≤i≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le i\\le 4$$\\end{document} and that A5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_5$$\\end{document} beats A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_1$$\\end{document} is 1/32+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/32+o(1)$$\\end{document}. Furthermore, the limiting tournamenton has range contained in the discrete set {0,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}$$\\end{document}. This proves that the associated tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of Cornacchia and Hązła regarding the continuous analogue of the balanced sequence model. The proof is based on a reduction to conditional central limit theorems (related to work of Polymath), the use of a “Poissonization” style method to reduce to computations with independent random variables, and the systematic use of switching-based arguments to extract cancellations in Fourier estimates when establishing local limit-type estimates.

Conditional central limit theorem for critical branching random walk

Conditional central limit theorem for critical branching random walk

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

Stein’s method for conditional central limit theorem

In the seventies, Charles Stein revolutionized the way of proving the central limit theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article we develop a novel approach, using Stein’s method for exchangeable pairs, to find a rate of convergence in the conditional central limit theorem of the form (Xn∣Yn=k), where (Xn,Yn) are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdős–Rényi random graph.

Conditional central limit theorem for subcritical branching random walk

Conditional central limit theorem for subcritical branching random walk

Prior latent distribution comparison for the RNN Variational Autoencoder in low-resource language modeling

Probabilistic Bayesian methods are widely used in the machine learning domain. Variational Autoencoder (VAE) is a common architecture for solving the Language Modeling task in a self-supervised way. VAE consists of a concept of latent variables inside the model. Latent variables are described as a random variable that is fit by the data. Up to now, in the majority of cases, latent variables are considered normally distributed. The normal distribution is a well-known distribution that can be easily included in any pipeline. Moreover, the normal distribution is a good choice when the Central Limit Theorem (CLT) holds. It makes it effective when one is working with i.i.d. (independent and identically distributed) random variables. However, the conditions of CLT in Natural Language Processing are not easy to check. So, the choice of distribution family is unclear in the domain. This paper studies the priors selection impact of continuous distributions in the Low-Resource Language Modeling task with VAE. The experiment shows that there is a statistical difference between the different priors in the encoder-decoder architecture. We showed that family distribution hyperparameter is important in the Low-Resource Language Modeling task and should be considered for the model training.

Statistical properties of sketching algorithms.

SummarySketching is a probabilistic data compression technique that has been largely developed by the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a smaller surrogate dataset. Typically, inference proceeds on the compressed dataset. Sketching algorithms generally use random projections to compress the original dataset, and this stochastic generation process makes them amenable to statistical analysis. We argue that the sketched data can be modelled as a random sample, thus placing this family of data compression methods firmly within an inferential framework. In particular, we focus on the Gaussian, Hadamard and Clarkson–Woodruff sketches and their use in single-pass sketching algorithms for linear regression with huge samples. We explore the statistical properties of sketched regression algorithms and derive new distributional results for a large class of sketching estimators. A key result is a conditional central limit theorem for data-oblivious sketches. An important finding is that the best choice of sketching algorithm in terms of mean squared error is related to the signal-to-noise ratio in the source dataset. Finally, we demonstrate the theory and the limits of its applicability on two datasets.

Rates of Convergence in Laplace’s Integrals and Sums and Conditional Central Limit Theorems

We obtained the exact estimates for the error terms in Laplace’s integrals and sums implying the corresponding estimates for the related laws of large number and central limit theorems including the large deviations approximation.

On confidence intervals for failure probability estimates in Kriging-based reliability analysis

Despite recent advancements in adaptive Kriging-based reliability analysis for complex limit states, estimation of the accuracy of extant techniques when the true failure probability is unknown remains an important challenge. The present study addresses this gap by developing analytical confidence intervals (CIs) for failure probability estimates. This is facilitated here by leveraging statistical properties of Poisson Binomial distribution for the expected number of failure points in the set of candidate design samples in adaptive Kriging as well as Lindeberg's condition for central limit theorem. Concerning computational demands involved in the computation of CIs, a simpler case where Kriging correlations are neglected is also derived. The performance of the proposed CIs is subsequently analyzed for five examples with different and varying complexities. Results indicate that the proposed CI with correlations considered offers the most accurate intervals. Additionally, whereas the CI estimated without Kriging correlation is not entirely satisfactory at early-stages of adaptive reliability analysis, it converges to accurate bounds at later stages.

Moments of the error term in the Sato–Tate law for elliptic curves

We derive new bounds for moments of the error in the Sato–Tate law over families of elliptic curves. Our estimates are stronger than those obtained in [4] and [5] for the first and second moment, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha [21] who derived a Central Limit Theorem on the distribution of the errors in the Sato–Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse–Weil L-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature.

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.

Fluctuation of Matrix Entries and Application to Outliers of Elliptic Matrices

AbstractFor any family of N ⨯ N randommatrices that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type Tr(AkM), where the matrix M is deterministic (such random variables include, for example, the normalized matrix entries of Ak). A consequence is the asymptotic independence of the projection of the matrices Ak onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Formof the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other.These phenomena have already been observed with random matrices from the Single Ring Theorem.

Conditional Central Limit Theorem

The arrays with rows consisting of conditionally independent random variables with respect to certain $\sigma$-algebras are studied. An analogue of the Lindeberg--Feller theorem known for systems of independent random variables is established. This result is based on the theorem proved by Yuan, Wei, and Lei in [J. Korean Math. Soc., 51 (2014), pp. 1--15], where the authors considered a sequence of random variables conditionally independent with respect to a given $\sigma$-algebra. They were interested in a.s. convergence, whereas our version of the Lindeberg condition in a weak form (involving convergence in probability) is less restrictive. An application of the indicated new result for arrays provides an extension of conditions for asymptotic normality of the estimates of the regression function second moment obtained in a recent paper by Gyorfi and Walk [J. Mach. Learn. Res., 16 (2015), pp. 1863--1877].

Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence $Y_{n}:=B_{1}+A_{1}B_{2}+\cdots+(A_{1}\cdots A_{n-1})B_{n}$, where $(A_{i},B_{i})\subset(0,\infty)\times\mathbb{R}$. Estimates for the stationary tail distribution of $\{Y_{n}\}$ have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if $M:=\sup_{n}Y_{n}$, then $\mathbb{P}\{M>u\}\sim\mathcal{C}_{M}u^{-\xi}$ as $u\to\infty$. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time $T_{u}:=(\log u)^{-1}\inf\{n:Y_{n}>u\}$. We begin by showing that, conditional on $\{T_{u}<\infty\}$, $T_{u}\to\rho$ as $u\to\infty$ for a certain positive constant $\rho$. We then provide a conditional central limit theorem for $\{T_{u}\}$, and study $\mathbb{P}\{T_{u}\in G\}$ as $u\to\infty$ for sets $G\subset[0,\infty)$. If $G\subset[0,\rho)$, then we show that $\mathbb{P}\{T_{u}\in G\}u^{I(G)}\to C(G)$ as $u\to\infty$ for a certain large deviation rate function $I$ and constant $C(G)$. On the other hand, if $G\subset(\rho,\infty)$, then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process $M_{n}^{\ast}:=\max\{A_{n}M_{n-1}^{\ast}+B_{n},0\}$, $n\in\mathbb{Z}_{+}$. Using Siegmund duality, we relate the first passage times of $\{Y_{n}\}$ to the finite-time exceedance probabilities of $\{M_{n}^{\ast}\}$, yielding a new result concerning the convergence of $\{M_{n}^{\ast}\}$ to its stationary distribution.

Limit theorems in the imitative monomer-dimer mean-field model via Stein’s method

We consider the imitative monomer-dimer model on the complete graph introduced in the work of Alberici et al. [J. Math. Phys. 55, 063301-1–063301-27 (2014)]. It was shown that this model is described by the monomer density and has a phase transition along certain coexistence curve, where the monomer and dimer phases coexist. More recently, it was understood [D. Alberici et al., Commun. Math. Phys. (published online, 2016)] that the monomer density exhibits the central limit theorem away from the coexistence curve and enjoys a non-normal limit theorem at criticality with normalized exponent 3/4. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein’s method of exchangeable pairs. Our approach recovers what were established in the work of Alberici et al. [Commun. Math. Phys. (published online, 2016)] and furthermore allows to obtain the conditional central limit theorems along the coexistence curve. In all these results, the Berry-Esseen inequalities for the Kolmogorov-Smirnov distance are given.

A Conditional Approach to Panel Data Models with Common Shocks

This paper studies the effects of common shocks on the OLS estimators of the slopes’ parameters in linear panel data models. The shocks are assumed to affect both the errors and some of the explanatory variables. In contrast to existing approaches, which rely on using results on martingale difference sequences, our method relies on conditional strong laws of large numbers and conditional central limit theorems for conditionally-heterogeneous random variables.

Some preliminary results on conditionally ψ-mixing sequences of random variables

From the ordinary notion of -mixing for a sequence of random variables, a new concept called conditionally -mixing is proposed. That conditionally -mixing neither implies nor is implied by -mixing is illustrated by examples. Certain criteria for checking conditionally -mixing property as well as basic properties are derived, and several conditional covariance inequalities are obtained. By means of these properties and inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established.

A quenched weak invariance principle

Dans cet article, nous étudions le théorème central limite conditionnel presque sûr, ainsi que sa forme fonctionnelle, pour des suites stationnaires de variables aléatoires réelles satisfaisant une condition de type projectif. Nous donnons des applications de ces résultats aux processus fortement mélangeants ainsi qu’à des chaînes de Markov nonirréductibles. Les preuves sont essentiellement basées sur une approximation normale de suites doublement indexées de variables aléatoires de type martingale.

SOME RESULTS ON CONDITIONALLY UNIFORMLY STRONG MIXING SEQUENCES OF RANDOM VARIABLES

From the ordinary notion of uniformly strong mixing for a se- quence of random variables, a new concept called conditionally uniformly strong mixing is proposed and the relation between uniformly strong mix- ing and conditionally uniformly strong mixing is answered by examples, that is, uniformly strong mixing neither implies nor is implied by condi- tionally uniformly strong mixing. A couple of equivalent definitions and some of basic properties of conditionally uniformly strong mixing ran- dom variables are derived, and several conditional covariance inequalities are obtained. By means of these properties and conditional covariance inequalities, a conditional central limit theorem stated in terms of condi- tional characteristic functions is established, which is a conditional version of the earlier result under the non-conditional case.

CONDITIONAL CENTRAL LIMIT THEOREMS FOR A SEQUENCE OF CONDITIONAL INDEPENDENT RANDOM VARIABLES

A conditional version of the classical central limit theorem is derived rigorously by using conditional characteristic functions, and a more general version of conditional central limit theorem for the case of conditionally independent but not necessarily conditionally identically distributed random variables is established. These are done anticipating that the field of conditional limit theory will prove to be of significant applicability.