Limit Theorem Research Articles

Equilibrium States of Endomorphisms of $\mathbb{P}^{k}$: Spectral Stability and Limit Theorems

AbstractWe establish the existence of a spectral gap for the transfer operator induced on $\mathbb{P}^{k} = \mathbb{P}^{k} (\mathbb{C})$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Threshold network GARCH model

Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its variations have been widely adopted in the study of financial volatilities, while the extension of GARCH‐type models to high‐dimensional data is always difficult because of over‐parameterization and computational complexity. In this article, we propose a multi‐variate GARCH‐type model that can simplify the parameterization by utilizing the network structure that can be appropriately specified for certain types of high‐dimensional data. The asymmetry in the dynamics of volatilities is also considered as our model adopts a threshold structure. To enable our model to handle data with extremely high dimension, we investigate the near‐epoch dependence (NED) of our model, and the asymptotic properties of our quasi‐maximum‐likelihood‐estimator (QMLE) are derived from the limit theorems for NED random fields. Simulations are conducted to test our theoretical results. At last we fit our model to log‐returns of four groups of stocks and the results indicate that bad news is not necessarily more influential on volatility if the network effects are considered.

A Conditioned Local Limit Theorem for Nonnegative Random Matrices

A Conditioned Local Limit Theorem for Nonnegative Random Matrices

New Combinatorial Identity for the Set of Partitions and Limit Theorems in Finite Free Probability Theory

Abstract We provide a refined combinatorial identity for the set of partitions of $\{1,\dots , n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida’s limit theorem. That is, we provide the limit of $\{D_{1/m}((p_{d}^{\boxtimes _{d}m})^{\boxplus _{d}m})\}_{m\in{\mathbb{N}}}$ as $m\rightarrow \infty $ in two cases: (i) $m/d\rightarrow t$ for some $t>0$ or (ii) $m/d\rightarrow 0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko’s limit theorems concerning the unitary Hermite and the Laguerre polynomials.

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three-variate low-order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID-19 pandemic in 2020.

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Explicit constants in the nonuniform local limit theorem for Poisson binomial random variables

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P(S=k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$P(S=k)$\\end{document} when S=∑i=1nXi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$S=\\sum_{i=1}^{n}X_{i}$\\end{document} and X1,X2,…,Xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X_{1},X_{2},\\ldots ,X_{n}$\\end{document} are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.

Probabilistic Stirling numbers and applications

AbstractWe introduce probabilistic Stirling numbers of the first kind $$s_Y(n,k)$$ s Y ( n , k ) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind $$S_Y(n,k)$$ S Y ( n , k ) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between $$s_Y(n,k)$$ s Y ( n , k ) and $$S_Y(n,k)$$ S Y ( n , k ) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning.

Increasing subsequences of linear size in random permutations and the Robinson–Schensted tableaux of permutons

AbstractThe study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson–Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size behaves as . Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the “RS‐tableaux” of a permuton are non‐trivial, we show that the RS‐tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of . We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS‐tableaux of a permuton satisfy a partial differential equation.

New copula families and mixing properties

New copula families and mixing properties

Inference for calendar effects in microstructure noise

We develop a statistical inference procedure for the ubiquitous calendar effects in microstructure noise using high frequency data. This is, to the best of our knowledge, the first inference theory ever built for noise calendar effect under the general semi‐martingale‐plus‐noise setup for prices contaminated with non‐stationary, endogenous, and serially dependent microstructure noise. We devise a noise‐calendar‐effect estimator by an appropriately scaled average of high‐frequency returns that precede a time of day across a large number of trading days. Feasible central limit theorem for the estimator is established under a joint infill and long‐span asymptotics. Monte Carlo simulations corroborate our theoretical findings. An empirical study on the high‐frequency data of the e‐mini S&P 500 futures and a Chinese stock demonstrates that the noise calendar effect has undergone significant changes over time for the latter, yet remains stable for the former.

Central Limit Theorems via Analytic Combinatorics in Several Variables

The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of analytic combinatorics in several variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. In this work we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish an explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem stated as a conjecture in recent work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. It is our hope that this work provides readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields.

The distribution of argmaximum or a winner problem

We consider a limit theorem for the distribution of a random variable (r.v.) An=argmaxi:1…n{Xi}, where Xi’s are independent continuous non-negative random r.v.’s. The Xi,i=1.…,n, may be interpreted as the gains of n players in a game, and the r.v. An itself as the number of a “winner”. The paper contains some limit theorems for the distribution of An as n→∞.

Analytical modeling of the spray amplification of a spatially smoothed laser beam

Spatial amplification of the near-forward Brillouin scattering (FSBS) produced by a laser beam smoothed with a random phase plate (RPP) is considered by using a novel technique based on the central limit theorem [C. Ruyer et al., Phys. Rev. E 107, 035208 (2023)]. It is demonstrated that FSBS amplification proceeds over a length much larger than the longitudinal speckle correlation length and, under certain conditions, scales as a square of the average gain coefficient. Analytical expressions for the spatial gain are successfully compared with paraxial electromagnetic simulations, demonstrating that the beamlet correlation through ion-acoustic waves dominates the spatial growth for intense enough laser beams. The scattered wave aperture increases with the gain and can extend beyond the small angle scattering limit. These results open the way for developing reduced modeling of beam spray amplification in radiation hydrodynamics codes.

A central limit theorem for the number of excursion set components of Gaussian fields

A central limit theorem for the number of excursion set components of Gaussian fields

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Limit Theorems for Self-Intersecting Trajectories in $$\mathbb {Z}$$-Extensions

Limit Theorems for Self-Intersecting Trajectories in $$\mathbb {Z}$$-Extensions

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

The functional central limit theorem for Markov-switching GARCH model

In this paper we consider the Markov Switching GARCH model suggested by Haas, Mittnik, and Paolella(2004). We show under proper assumptions that the functional central limit theorems hold for the process, the square of the process, and regime variances. The functional central limit theorem for a linear combination of regime variances is also obtained.