Multidimensional Central Limit Theorem Research Articles

Quantitative two-scale stabilization on the Poisson space

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one—that we evaluate and compare at two different scales—and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization—see Penrose and Yukich (Ann. Appl. Probab. 11 (2001) 1005–1041) and Penrose (Ann. Probab. 33 (2005) 1945–1991). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (Ann. Appl. Probab. 27 (2017) 1588–1645) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbour graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

Functional convergence of sequential U-processes with size-dependent kernels

We consider sequences of $U$-processes based on symmetric kernels of a fixed order, that possibly depend on the sample size. Our main contribution is the derivation of a set of analytic sufficient conditions, under which the aforementioned $U$-processes weakly converge to a linear combination of time-changed independent Brownian motions. In view of the underlying symmetric structure, the involved time-changes and weights remarkably depend only on the order of the U-statistic, and have consequently a universal nature. Checking these sufficient conditions requires calculations that have roughly the same complexity as those involved in the computation of fourth moments and cumulants. As such, when applied to the degenerate case, our findings are infinite-dimensional extensions of the central limit theorems (CLTs) proved in de Jong (1990) and D\"obler and Peccati (2017). As important tools in our analysis, we exploit the multidimensional central limit theorems established in D\"obler and Peccati (2019) together with upper bounds on absolute moments of degenerate $U$-statistics by Ibragimov and Sharakhmetov (2002), and also prove some novel multiplication formulae for degenerate symmetric $U$-statistics -- allowing for different sample sizes -- that are of independent interest. We provide applications to random geometric graphs and to a class of $U$-statistics of order two, whose Gaussian fluctuations have been recently studied by Robins et al. (2016), in connection with quadratic estimators in non-parametric models. In particular, our application to random graphs yields a class of new functional central limit theorems for subgraph counting statistics, extending previous findings in the literature. Finally, some connections with invariance principles in changepoint analysis are established.

On intermediate levels of a nested occupancy scheme in a random environment generated by stick-breaking II

A nested occupancy scheme in a random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in a random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper, we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that n balls have been thrown, denote by the number of occupied boxes in the jth level and call the level j intermediate if and as . We prove a multidimensional central limit theorem for the vector , properly normalized and centred, as , where and . The present paper continues the line of investigation initiated in the article [D. Buraczewski, B. Dovgay, and A. Iksanov, On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I, Electron. J. Probab. 25(123) (2020), pp. 1–24] in which the occupancy of intermediate levels , was analysed.

Central limit theorem for the least common multiple of a uniformly sampled m-tuple of integers

Let Bn(m) be a set picked uniformly at random among all m-elements subsets of {1,2,…,n}. We provide a pathwise construction of the collection (Bn(m))1⩽m⩽n and prove that the logarithm of the least common multiple of the integers in (Bn(⌊mt⌋))t⩾0, properly centered and normalized, converges to a Brownian motion when both m,n tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of m independent random variables having uniform distribution on {1,2,…,n}. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.

On the Achievable Rate Under Finite Codelength for RIS-Aided Systems

The ideal Shannon channel capacity can only be achieved by infinite codelength, so it is not accurate for practical finite codelength. The codelength of channel coding schemes heavily affects the performance of communication systems, but it cannot be too long to ensure affordable decoding complexity and latency. Unfortunately, existing performance analysis for systems aided by reconfigurable intelligent surface (RIS) only focus on the ideal channel capacity and neglect the achievable rate loss caused by finite codelength. To fill in this gap, this paper analyzes the achievable rate under finite codelength in a time-coherent RIS-aided systems, and gives an approximate analytic solution based on Taylor expansion and multi-dimensional central limit theorem. To be more specific, the closed form approximated achievable rate is given based on Taylor expansion, and it is shown that the required codelength decreases superlinearly with the increase of the number of RIS elements. Simulation results verify the analysis, and demonstrate the advantage of RIS-aided systems over the traditional large antenna array schemes.

Statistical limit laws for hyperbolic groups

Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text{CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.

Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem

We use Stein’s method to bound the Wasserstein distance of order 2 between a measure $$\nu $$ and the Gaussian measure using a stochastic process $$(X_t)_{t \ge 0}$$ such that $$X_t$$ is drawn from $$\nu $$ for any $$t > 0$$ . If the stochastic process $$(X_t)_{t \ge 0}$$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $$p \ge 1$$ . Using our results, we provide convergence rates for the multi-dimensional central limit theorem in terms of Wasserstein distances of any order $$p \ge 2$$ under simple moment assumptions.

On Asymptotic Expansions in the “Interval” CLT for Sums of Independent Random Vectors

We study the remainder term taking into account the asymptotic expansions in the multidimensional central limit theorem for sums of independent random vectors. The dependence of the remainder term on the measure of hitting set is studied.

On Integro-Local CLT for Sums of Independent Random Vectors

The remainder term in the integro-local version of the multidimensional central limit theorem for a sum of independent random vectors is studied with due account of asymptotic expansions. It is assumed that the distribution of this sum can be absolutely continuous and/or lattice in some coordinates.

Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles

We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.

Quantitative normal approximation of linear statistics of $\beta$-ensembles

We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.

From the master equation to mean field game limit theory: a central limit theorem

Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.

A Functional Limit Theorem for the Sine-Process

The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the Kolmogorov property and satisfies the Central Limit Theorem, but our functional limit theorem is very different from the Donsker Invariance Principle. We show that the time integral of our process can be approximated by the sum of a linear Gaussian process and independent Gaussian fluctuations whose covariance matrix is computed explicitly. We interpret these results in terms of the Gaussian Free Field convergence for the random matrix models. The proof relies on a general form of the multidimensional Central Limit Theorem under the sine-process for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity $1/2$.

A new central limit theorem and decomposition for Gaussian polynomials, with an application to deterministic approximate counting

One of the main results of this paper is a new multidimensional central limit theorem (CLT) for multivariate polynomials under Gaussian inputs. Roughly speaking, the new CLT shows that any collection of Gaussian polynomials with small eigenvalues (suitably defined) must have a joint distribution which is close to a multidimensional Gaussian distribution. The new CLT is proved using tools from Malliavin calculus and Stein’s method. A second main result of the paper, which complements the new CLT, is a new decomposition theorem for low-degree multilinear polynomials over Gaussian inputs. Roughly speaking, this result shows that (up to some small error) any such polynomial is very close to a polynomial which can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. An important feature of this decomposition theorem is the delicate control obtained between the number of polynomials in the decomposition versus their eigenvalues. As the main application of these results, we give a deterministic algorithm for approximately counting satisfying assignments of a degree-d polynomial threshold function (PTF) over the domain $$\{-1,1\}^n$$ ; this is a well-studied problem from theoretical computer science. More precisely, given as input a degree-d polynomial $$p(x_1,\dots ,x_n)$$ over $${{\mathbb {R}}}^n$$ and a parameter $$\epsilon > 0$$ , the algorithm approximates $$\begin{aligned} \mathop {\mathbf{Pr}}_{x \sim \{-1,1\}^n}[p(x) \ge 0] \end{aligned}$$ to within an additive $$\pm \epsilon $$ in time $$O_{d,\epsilon }(1)\cdot \mathrm {poly}(n^d)$$ . (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of the algorithm (as a function of $$n^d$$ , the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem (Kane, CoRR, arXiv:1210.1280 , 2012), based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time $$n^{O_{d,c}(1) \cdot \epsilon ^{-c}}$$ for all $$c > 0$$ .

A note on a local limit theorem for Wiener space valued random variables

We prove a local limit theorem, that is, a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein–Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired $\mathcal{L}^{1}$-convergence of the density of $\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$. We close the paper comparing our result with certain Berry–Esseen bounds for multidimensional central limit theorems.

Multidimensional central limit theorem for sums of functions of the trajectories of endomorphisms

We study the rate of convergence in the central limit theorem for vector-valued sequences generated by endomorphisms of a multidimensional torus. In the proved theorem for sums of functions of the trajectories of endomorphisms of s-dimensional Euclidean space it is obtained almost optimal rate of convergence to the normal distribution. In the proof we use “method of successive approximations”, developed by us earlier (see Dubrovin V.T., Moskvin D.A. Theory of Probability & Its Applications, 1980, V. 24, Is. 3, P. 560–571) to prove limit theorems taking into account the rate of convergence for the sums of functions of sequences that satisfy a mixing condition.

Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the $$\beta $$ -Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.

Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I

We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas ensemble with an external potential V = q2mx2m. More precisely, let Pn(dH) = Cne-nTrV(H)dH be the distribution of n × n Hermitian random matrices, ρV (x)dx the equilibrium measure, where Cn is a normalization constant, V (x) = q2mx2m with \(q2m = \frac{{\Gamma \left( m \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {\frac{{2m + 1}}{2}} \right)}}\), and m ≥ 1. Let x1 ≤... ≤ xn be the eigenvalues of H. Let k:= k(n) be such that \(\frac{{k\left( n \right)}}{n} \in \left[ {a,1 - a} \right]\) for n large enough, where a ∈ (0, 1/2). Define \(G\left( s \right): = \int_{ - 1}^s {\rho v\left( x \right)dx, - 1 \leqslant s \leqslant 1} ,\) and set t:= G−1(k/n). We prove that, as n → ∞, \(\frac{{xk - t}}{{\frac{{\left( {\sqrt {\log n} } \right)}}{{\sqrt {2{\pi ^2}} n\rho v\left( t \right)}}}} \to N\left( {0,1} \right)\) in distribution. Multi-dimensional central limit theorem is also proved. Our results can be viewed as natural extensions of the bulk central limit theorems for GUE ensemble established by J. Gustavsson in 2005.

Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.

Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

A new nonparametric estimator of the local Hurst function of a multifractional Gaussian process based on the increment ratio (IR) statistic is defined. In a general frame, the point-wise and uniform weak and strong consistency and a multidimensional central limit theorem for this estimator are established. Similar results are obtained for a refinement of the generalized quadratic variations (QV) estimator. The example of the multifractional Brownian motion is studied in detail. A simulation study is included showing that the IR-estimator is more accurate than the QV-estimator.