Central Limit Theorem Research Articles

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.

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

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.

Identifying the Effect of Cognitive Motivation with the Method Based on Temporal Association Rule Mining Concept

Being motivated has positive influences on task performance. However, motivation could result from various motives that affect different parts of the brain. Analyzing the motivation effect from all affected areas requires a high number of EEG electrodes, resulting in high cost, inflexibility, and burden to users. In various real-world applications, only the motivation effect is required for performance evaluation regardless of the motive. Analyzing the relationships between the motivation-affected brain areas associated with the task’s performance could limit the required electrodes. This study introduced a method to identify the cognitive motivation effect with a reduced number of EEG electrodes. The temporal association rule mining (TARM) concept was used to analyze the relationships between attention and memorization brain areas under the effect of motivation from the cognitive motivation task. For accuracy improvement, the artificial bee colony (ABC) algorithm was applied with the central limit theorem (CLT) concept to optimize the TARM parameters. From the results, our method can identify the motivation effect with only FCz and P3 electrodes, with 74.5% classification accuracy on average with individual tests.

Estimates of constants in the limit theorems for chaotic dynamical systems

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

Central limit theorem for components in meandric systems through high moments

Abstract We investigate here the behaviour of a large typical meandric system, proving a central limit theorem for the number of components of a given shape. Our main tool is a theorem of Gao and Wormald that allows us to deduce a central limit theorem from the asymptotics of large moments of our quantities of interest.

CLT for $$\beta $$-Ensembles at High Temperature and for Integrable Systems: A Transfer Operator Approach

AbstractIn this paper, we prove a polynomial central limit theorem for several integrable models and for the $$\beta $$ β -ensembles at high temperature with polynomial potential. Furthermore, we connect the mean values, the variances and the correlations of the moments of the Lax matrices of these integrable systems with the ones of the $$\beta $$ β -ensembles. Moreover, we show that the local functions’ space-correlations decay exponentially fast for the considered integrable systems. For these models, we also established a Berry–Esseen-type bound.

Occupation times for age-structured branching processes

We consider occupation time processes for the age-structured branching processes with or without immigration introduced by Ji and Li (2022). Two large number laws and a central limit theorem for occupation times of the model with immigration are obtained.

Central limit theorem for the number of records in Fα-scheme

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.

Functional central limit theorems for rough volatility

AbstractThe non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent $H \in (0,1)$ H ∈ ( 0 , 1 ) . Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of $H$ H ). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Separable sample covariance matrices under elliptical populations with applications

This paper is to investigate the spectral properties of separable covariance matrices under elliptical populations. The separable covariance matrix model can handle both cross-row and cross-column correlations thus gain more popularity recently. Under the high-dimensional setting where the dimension p p and the sample size n n tend to infinity proportionally, we find the limit of the empirical spectral distribution and establish the central limit theorems (CLT) for linear spectral statistics of such kinds of sample covariance matrices. Some applications of our established CLT are also given.

A Deterministic Chaos-Model-Based Gaussian Noise Generator

The abilities of quantitative description of noise are restricted due to its origin, and only statistical and spectral analysis methods can be applied, while an exact time evolution cannot be defined or predicted. This emphasizes the challenges faced in many applications, including communication systems, where noise can play, on the one hand, a vital role in impacting the signal-to-noise ratio, but possesses, on the other hand, unique properties such as an infinite entropy (infinite information capacity), an exponentially decaying correlation function, and so on. Despite the deterministic nature of chaotic systems, the predictability of chaotic signals is limited for a short time window, putting them close to random noise. In this article, we propose and experimentally verify an approach to achieve Gaussian-distributed chaotic signals by processing the outputs of chaotic systems. The mathematical criterion on which the main idea of this study is based on is the central limit theorem, which states that the sum of a large number of independent random variables with similar variances approaches a Gaussian distribution. This study involves more than 40 mostly three-dimensional continuous-time chaotic systems (Chua’s, Lorenz’s, Sprott’s, memristor-based, etc.), whose output signals are analyzed according to criteria that encompass the probability density functions of the chaotic signal itself, its envelope, and its phase and statistical and entropy-based metrics such as skewness, kurtosis, and entropy power. We found that two chaotic signals of Chua’s and Lorenz’s systems exhibited superior performance across the chosen metrics. Furthermore, our focus extended to determining the minimum number of independent chaotic signals necessary to yield a Gaussian-distributed combined signal. Thus, a statistical-characteristic-based algorithm, which includes a series of tests, was developed for a Gaussian-like signal assessment. Following the algorithm, the analytic and experimental results indicate that the sum of at least three non-Gaussian chaotic signals closely approximates a Gaussian distribution. This allows for the generation of reproducible Gaussian-distributed deterministic chaos by modeling simple chaotic systems.

On the Local Central Limit Theorem for Interacting Spin Systems

On the Local Central Limit Theorem for Interacting Spin Systems

Fifteen Years of Experience with Surgical Management of Bilateral Non-familial Carotid Body Tumors

Bilateral carotid body tumors (CBTs) clinical manifestation is infrequent. We conducted this work to describe our experience in the surgical treatment of bilateral CBT and to analyze our results. A retrospective, observational study. We analyzed the totality of bilateral CBT resections that had been performed in our institution from January 2008 to September 2023. Data was obtained from medical records and anonymized, ethics approval was obtained from our institution committee. As the number of observations was less than those required by the central limit theorem our sample was considered nonparametric. Statistical analysis was performed on Stata17. We evaluated 16 patients with a total of 32 CBT; surgical resection was performed in 28 cases (87.50%). Median age of the patients was 60years (interquartile range [IQR] 46-64). Regarding the Shamblin classification, 9 CBTs (32.14%) were classified as Shamblin I, 11 (39.29%) as Shamblin II, and 8 (28.57%) as Shamblin III. The median Distance to the Base of the Skull (DTBOS) was 3.5cm (IQR 2.7-5.1), and the median tumor volume was 11.25cc (IQR 3.4-18.7). The median bleeding volume was 300ml (IQR 200-500), and the median surgical time was 190min (IQR 145-240). All surgeries were performed using the Retrocarotid Dissection technique. We documented 9 (32.14%) cases of nerve injuries, all of which were transitory. In the median regression a statistically significant association was found between DTBOS, Shamblin classification and tumor volume with intraoperative bleeding and length of stay. Surgical treatment remains safe and should be considered the gold standard for accurate histologic diagnosis. DTBOS and tumor volume, in addition to Shamblin classification, must be considered in preoperative planning to predict bleeding and hospital stay.

Diophantine conditions in the law of the iterated logarithm for lacunary systems

AbstractIt is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory’s limit theorems may fail for lacunary sums $$\sum f(n_k x)$$ ∑ f ( n k x ) if the sequence $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 has a strong arithmetic “structure”. The presence of such structure can be assessed in terms of the number of solutions $$k,\ell $$ k , ℓ of two-term linear Diophantine equations $$a n_k - b n_\ell = c$$ a n k - b n ℓ = c . As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős–Fortet example, and guarantees that $$\sum f(n_k x)$$ ∑ f ( n k x ) satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure “truly independent” behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 to ensure that $$\sum f(n_k x)$$ ∑ f ( n k x ) shows “truly random” behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.

Linear statistics for Coulomb gases: higher order cumulants

We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature β. In the large N limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form LN=∑i=1Nf(xi) , where x i ’s are the positions of the particles and where f(xi) is a sufficiently regular function. There exists at present standard results for the first and second moments of LN in the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of LN at large N, when the function f(x)=f(|x|) and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f′(|x|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d = 2 at the special value β = 2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of LN .

Large-dimensional central limit theorem with fourth-moment error bounds on convex sets and balls

Large-dimensional central limit theorem with fourth-moment error bounds on convex sets and balls

A feasible central limit theorem for realised covariation of SPDEs in the context of functional data

A feasible central limit theorem for realised covariation of SPDEs in the context of functional data

Intraday volatility patterns from short-dated options

We propose a nonparametric estimator for the deterministic periodic component of volatility from short-dated options within an in-fill asymptotic setting. The estimator uses options with zero and one day to expiration sampled at high-frequency during a trading day. At each point in time, we aggregate the options to form nonparametric estimates of conditional risk-neutral expectations of future integrated return variation for the two available option tenors. A suitable ratio of these estimates removes the stochastic components of the conditional expectations of future volatility, up to asymptotically higher-order terms, and allows to form estimates of the deterministic periodic component of volatility. We derive a Central Limit Theorem for the estimator, with its rate of convergence determined from the mesh of the strike grid and the length of the time to expiration of the options. The newly-developed estimation procedure is applied to S&P 500 index options data.

원전의 안전계통 설정치 불확도 평가에서의 중심극한정리 타당성 확인

In evaluating the uncertainty of nuclear power plant safety system setpoints, the central limit theorem(CLT) is often used when combining the uncertainties of each element. However, depending on the distribution characteristics of individual elements and the number of elements to be combined, there are cases where the central limit theorem cannot be applied. International standards are being revised to reflect this. In this paper, we describe various methodologies for calculating safety system setpoints and confirm whether the combined components follow the central limit theorem according to the distribution characteristics of the individual elements. The Monte Carlo simulation method is used as a basic tool to confirm this, and normality is verified using KS test in Python program library. Additionally, the variance characteristics of each distribution are examined and differences from the uniform distribution are analyzed. As a result of the analysis, in order for the combined component to follow the central limit theorem, if the distribution is concentrated on both sides compared to the uniform distribution, the number of input variables needs to be four or more. That is, when the number of input element uncertainty of the safety system setpoint exceeds 4, the combined uncertainty was evaluated to have normality regardless of the distribution of input element uncertainty. Therefore, it was confirmed that the method of applying CLT to calculate the coverage factor and then using it to obtain the expanded uncertainty was valid. In the case of a distribution concentrated toward the center compared to a uniform distribution, the number of variables less than 3 was required. The U type distribution, which is used to evaluate the uncertainty of wireless communication, showed characteristics closer to a uniform distribution than an inverted triangular distribution. Also, in the case of variance, it was found that the variance of the uniform distribution was larger than that of the linear distribution, confirming that this conformed to the maximum entropy principle.