Martingale Central Limit Theorem Research Articles

Revisit of a Diaconis urn model

Let G be a finite Abelian group of order d. We consider an urn in which, initially, there are labeled balls that generate the group G. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees, 1994). Siegmund and Yakir (2005) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group G. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.

Tests in functional autoregressive processes via local asymptotic normality condition

Local asymptotic normality (LAN) and uniform local asymptotic normality (ULAN) conditions are obtained for a linear functional autoregressive process of order p, p≥1, using a martingale central limit theorem. As a consequence of the LAN property, asymptotically efficient tests for determining the order are constructed. An example is discussed as a special cases.

Diffusion approximation of an infinite-server queue under Markovian environment with rapid switching

We consider an infinite-server queue with Markov-modulated non-homogeneous arrival and service processes. We adopt the martingale central limit theorem to derive the diffusion approximation of the centered and normalized queue length processes of the queueing system under suitable scaling. In particular, the diffusion approximation results in an Ornstein–Uhlenbeck process with time-varying coefficients and the associated covariance captures the stochastic and predictable variabilities simultaneously.

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation h=logu, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution uε of SHE with the flat initial condition and with noise which is mollified in space on scale in ε and its strength is weakened as βε=βˆ 2π −logε, and they prove that when βˆ∈(0,1), 1 βε(loguε−E[loguε]) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation uε with a general initial condition u0 and its transformation F(uε) for F in a class of functions F, which contains F(x)=xp (0<p≤1) and F(x)=logx. Then, we prove that 1 βε(F(uε(t,⋅))−E[F(uε(t,⋅))]) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F∈F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.

On Bond Incident Degree Indices of Random Spiro Chains

Let denote a graph, many important topological indices can be defined as In this paper, we study these kinds of topological indices in random spiro chains via a martingale approach. In which their explicit analytical expressions of the exact distribution, expected value, and variance are obtained. As n goes to the asymptotic normality of topological indices of a random spiro chain is established through the Martingale Central Limit Theorem. In particular, we compute the Nirmala, Sombor, Randić, and Zagreb index for a random spiro chain along with their comparative analysis.

Extremal quantile autoregression for heavy-tailed time series

The conventional quantile estimator of the extreme conditional quantiles under quantile autoregression models is relatively unstable due to data sparsity in the tails. Extreme value theory provides a mathematical foundation of extrapolation to estimate extreme quantiles. However, the asymptotic distributions of existing estimators are often complicated making it inconvenient to apply for statistical inference. We develop a new adaptive estimation procedure based on a generalized Hill estimator of the extreme value index to estimate extreme conditional quantiles in autoregression models. We establish the asymptotic normality of the proposed estimators under some regularity conditions by applying the martingale central limit theorem. The asymptotic variances have closed expressions and can be directly estimated. Based on these results, we propose a procedure to construct confidence intervals for the extreme conditional quantiles. We also provide a diagnostic tool to check the heavy-tailedness of the distribution. Simulation studies and a real data analysis are carried out to demonstrate the advantages of the proposed methods over existing approaches.

Exponentially Weighted Multivariate HAR Model with Applications in the Stock Market

This paper considers a multivariate time series model for stock prices in the stock market. A multivariate heterogeneous autoregressive (HAR) model is adopted with exponentially decaying coefficients. This model is not only suitable for multivariate data with strong cross-correlation and long memory, but also represents a common structure of the joint data in terms of decay rates. Tests are proposed to identify the existence of the decay rates in the multivariate HAR model. The null limiting distributions are established as the standard Brownian bridge and are proven by means of a modified martingale central limit theorem. Simulation studies are conducted to assess the performance of tests and estimates. Empirical analysis with joint datasets of U.S. stock prices illustrates that the proposed model outperforms the conventional HAR models via OLSE and LASSO with respect to residual errors.

Distribution of tree parameters by martingale approach

AbstractFor a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem and the Aldous–Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.

A new test for high‐dimensional regression coefficients in partially linear models

Partially linear regression models are semiparametric models that contain both linear and nonlinear components. They are extensively used in many scientific fields for their flexibility and convenient interpretability. In such analyses, testing the significance of the regression coefficients in the linear component is typically a key focus. Under the high‐dimensional setting, i.e., “large p, small n,” the conventional F‐test strategy does not apply because the coefficients need to be estimated through regularization techniques. In this article, we develop a new test using a U‐statistic of order two, relying on a pseudo‐estimate of the nonlinear component from the classical kernel method. Using the martingale central limit theorem, we prove the asymptotic normality of the proposed test statistic under some regularity conditions. We further demonstrate our proposed test's finite‐sample performance by simulation studies and by analyzing some breast cancer gene expression data.

A faster U-statistic for testing independence in the functional linear models

Testing the dependence between the response and the functional predictor in a functional linear model is of fundamental importance. In this paper, based on a U-statistic of order two, we develop a computationally more efficient test for lacking of dependence in functional linear regression model. By the martingale central limit theorem, we prove that the asymptotic normality of the proposed test statistic under some mild regularity conditions. Simulation results show that our proposed test can be tens or hundreds time faster than the FLUTE test by Hu et al. (2020) which uses a U-statistic of order four. We further demonstrate the superiority of our test by two real data applications.

From Univariate to Multivariate Coupling Between Continuous Signals and Point Processes: A Mathematical Framework.

Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties of coupling measures between continuous signals and point processes. We first introduce martingale stochastic integration theory as a mathematical model for a family of statistical quantities that include the phase locking value, a classical coupling measure to characterize complex dynamics. Based on the martingale central limit theorem, we can then derive the asymptotic gaussian distribution of estimates of such coupling measure that can be exploited for statistical testing. Second, based on multivariate extensions of this result and random matrix theory, we establish a principled way to analyze the low-rank coupling between a large number of point processes and continuous signals. For a null hypothesis of no coupling, we establish sufficient conditions for the empirical distribution of squared singular values of the matrix to converge, as the number of measured signals increases, to the well-known Marchenko-Pastur (MP) law, and the largest squared singular value converges to the upper end of the MP support. This justifies a simple thresholding approach to assess the significance of multivariate coupling. Finally, we illustrate with simulations the relevance of our univariate and multivariate results in the context of neural time series, addressing how to reliably quantify the interplay between multichannel local field potential signals and the spiking activity of a large population of neurons.

Normal approximation and fourth moment theorems for monochromatic triangles

AbstractGiven a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well‐known fourth‐moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth‐moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].

Design and Analysis of Cancer Clinical Trials for Personalized Medicine.

Biomarkers play a key role in the development of personalized medicine. Cancer clinical trials with biomarker should be appropriately designed and analyzed reflecting the various factors, such as the phase of trials, the type of biomarker, the study objectives, and whether the used biomarker is already validated or not. In this paper, we demonstrate design and analysis of two phase II cancer clinical trials, one with a predictive biomarker and the other with a prognostic biomarker. A statistical testing method and its sample size calculation method are presented for each of the trials. We assume that the primary endpoint of these trials is a time to event variable, but this concept can be used for any type of endpoint with associated testing methods. The test statistics and their sample size formulas are derived using the large sample approximation based on the martingale central limit theorem. Using simulations, we find that the test statistics control the type I error rate accurately and the sample sizes calculated using the formulas maintain the statistical power specified at the design stage.

Interest rate risk in the banking book: A closed-form solution for non-maturity deposits

I present an analytical valuation framework for the management of fixed-income instruments traded in imperfectly competitive markets, like demand deposits and credit card loans in the banking book, inter alia, to stabilize the abnormal profit margin. Banking book instruments contain embedded options such as withdrawal rights, discretionary pricing, rate clustering and zero-based floors. Analytical solutions speed up computation time to calculate valuations, earnings and risk measures like closed-form expressions for margin spreads, hedge ratios and parameter sensitivities. Asymptotically, according to martingale central limit theorems and thanks to the long-term nature of the banking book, Gaussian approximations can be applied.

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

Renz [13] has established a rate of convergence 1/n in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen [2] and Grama and Haeusler [6], is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

A probabilistic approach to the [formula omitted]-variation of classical fractal functions with critical roughness

We consider Weierstraß and Takagi–van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing pth variation for all p>1 but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener–Young Φ-variation along the sequence of b-adic partitions, where Φ(x)=x∕−logx. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi–van der Waerden functions, we use the CLT for Markov chains if a certain parameter b is odd, and the standard CLT for b even.

On the Wasserstein distance for a martingale central limit theorem

We prove an upper bound on the Wasserstein distance between normalized martingales and the standard normal random variable, which extends a result of Röllin (2018). The proof is based on a method of Bolthausen (1982).

Statistical Inference for Online Decision Making: In a Contextual Bandit Setting

Online decision making problem requires us to make a sequence of decisions based on incremental information. Common solutions often need to learn a reward model of different actions given the contextual information and then maximize the long-term reward. It is meaningful to know if the posited model is reasonable and how the model performs in the asymptotic sense. We study this problem under the setup of the contextual bandit framework with a linear reward model. The ε-greedy policy is adopted to address the classic exploration-and-exploitation dilemma. Using the martingale central limit theorem, we show that the online ordinary least squares estimator of model parameters is asymptotically normal. When the linear model is misspecified, we propose the online weighted least squares estimator using the inverse propensity score weighting and also establish its asymptotic normality. Based on the properties of the parameter estimators, we further show that the in-sample inverse propensity weighted value estimator is asymptotically normal. We illustrate our results using simulations and an application to a news article recommendation dataset from Yahoo!. Supplementary materials for this article are available online.

Sparse Network Asymptotics for Logistic Regression

Consider a bipartite network where N consumers choose to buy or not to buy M different products. This paper considers the properties of the logistic regression of the N × M array of “i-buys-j” purchase decisions, [Y ij ] 1≤i≤N,≤j≤M , onto known functions of consumer and product attributes under asymptotic sequences where (i) both N and M grow large and (ii) the average number of products purchased per consumer is finite in the limit. This latter assumption implies that the network of purchases is sparse: only a (very) small fraction of all possible purchases are actually made (concordant with many real-world settings). Under sparse network asymptotics, the first and last terms in an extended Hoeffding-type variance decomposition of the score of the logit composite log-likelihood are of equal order. In contrast, under dense network asymptotics, the last term is asymptotically negligible. Asymptotic normality of the logistic regression coefficients is shown using a martingale central limit theorem (CLT) for triangular arrays. Unlike in the dense case, the normality result derived here also holds under degeneracy of the network graphon. Relatedly, when there “happens to be” no dyadic dependence in the dataset in hand, it specializes to recently derived results on the behavior of logistic regression with rare events and iid data. Sparse network asymptotics may lead to better inference in practice since they suggest variance estimators which (i) incorporate additional sources of sampling variation and (ii) are valid under varying degrees of dyadic dependence.

Testing independence in high-dimensional multivariate normal data

A simple statistic was proposed for testing the complete independence of components of random vector having a multivariate normal distribution. Using the martingale central limit theorem, it was proved that this test statistic converges in distribution to the normal as long as both the sample size and dimension go to infinity. Simulations were also carried out to performance evaluation of the proposed test for dealing with the high-dimensional data and the results were compared to those of the state-of-the-art tests. The results confirmed that, in terms of the average relative error criterion, the proposed test has adequate type I error rate control and comparable power to those tests with smaller size distortions.