We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. A possible application -- a CLT for the hitting time for random walk on random graphs -- will be presented in \cite{LoTe20b}
We prove weak convergence of triangular arrays to the compound Poisson limit using Tikhomirov's method. The result is applied to statistical estimation of the threshold parameter in autoregressive models.
In this paper, we establish the first and the second-order asymptotics of distributions of normalized maxima of independent and non-identically distributed bivariate Gaussian triangular arrays, where each vector of the $n$th row follows from a bivariate Gaussian distribution with correlation coefficient being a monotone continuous function of $i/n$. Furthermore, parametric inference for this unknown function is studied. Some simulation study and real data sets analysis are also presented.
In previous papers, we studied the asymptotic behaviour of $S_N(A,X)=(2N+1)^{-d/2}\sum_{n \in A_N} X_n,$ where $X$ is a centered, stationary and weakly dependent random field, and $A_N=A \cap [-N,N]^d$, $A \subset \mathbb{Z}^d$. This leads to the definition of asymptotically measurable sets, which enjoy the property that $S_N(A;X)$ has a Gaussian weak limit for any $X$ belonging to a certain class. Here we extend this type of results to the case of weakly dependent triangular arrays and present an application of this technique to regression models. Indeed, we prove that CLT and related results…
Study of time series data often involves measuring the strength of temporal dependence, on which statistical properties like consistency and central limit theorem are built. Historically, various dependence measures have been proposed. In this note, we first survey some of the most well-used dependence measures as well as various probability and moment inequalities built upon them under a high-dimensional triangular array time series setting. We then argue that this triangular array setting will pose substantially new challenges to the verification of some dependence conditions. In particular,…
Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation.
A general moment bound for sums of products of Gaussian vector's functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1981), Arcones (1994) and others. A Berry-Esseen-type bound in the above-mentioned central limit theorem is derived following Nourdin, Peccati and Podolskij (2011). Two applications of the above results are discussed. The first one refers to the as…
Inference about dependencies in a multiway data array can be made using the array normal model, which corresponds to the class of multivariate normal distributions with separable covariance matrices. Maximum likelihood and Bayesian methods for inference in the array normal model have appeared in the literature, but there have not been any results concerning the optimality properties of such estimators. In this article, we obtain results for the array normal model that are analogous to some classical results concerning covariance estimation for the multivariate normal model. We show that under …
We establish the upper bound on the speed of convergence to the infinitely divisible limit density in the local limit theorem for triangular arrays of random variables $\{X_{k,n},\, k=1,..,a_n, \, n\in \nat\}$.
In this paper, we study second order expansions of distributions of maxima of bivariate Gaussian triangular arrays under power normalization. Numerical analysis are given to compare the asymptotic behaviors under power normalization with the asymptotic behaviors under linear normalization derived by Hashorva et al. (2016).
In this paper we investigate the asymptotic behaviour of the componentwise maxima for two bivariate skew elliptical triangular arrays with components given in terms of skew transformations of bivariate spherical random vectors. We find the weak limit of the normalized maxima for both cases that the random radius pertaining to the elliptical random vectors is either in the Gumbel or in the Weibull max-domain of attractions.
In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of $i$th vector of given $n$th row is the function of $i/n$. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.
We prove a martingale triangular array generalization of the Chow-Birnbaum-Marshall's inequality. The result is used to derive a strong law of large numbers for martingale triangular arrays whose rows are asymptotically stable in a certain sense. To illustrate, we derive a simple proof, based on martingale arguments, of the consistency of kernel regression with dependent data. Another application can be found in \cite{atchadeetfort08} where the new inequality is used to prove a strong law of large numbers for adaptive Markov Chain Monte Carlo methods.
In this paper we obtain the central limit theorem for triangular arrays of non-homogeneous Markov chains under a condition imposed to the maximal coefficient of correlation. The proofs are based on martingale techniques and a sharp lower bound estimate for the variance of partial sums. The results complement an important central limit theorem of Dobrushin based on the contraction coefficient.
It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.
Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables.
In \cite{MPU} a functional CLT was obtained for triangular arrays satisfying the Lindeberg condition, that the sum of the individual variances is at most the same order as the variance of the underlying sum, and under the optimal mixing rats $\sum_{n}ρ(2^n)<\infty$, where $ρ(\cdot)$ are the $ρ$-mixing coefficients of the array. In this paper we will present alternative conditions which do not involve the assumption on the sum of variances, and instead we will assume certain maximal moment assumptions (which we can verify for $ϕ$-mixing arrays) and mixing rates of the form $\sum_nρ(e^{G(n)})<\i…
In this paper we continue our study of exit times for random walks with independent but not necessarily identical distributed increments. Our paper "First-passage times for random walks with non-identically distributed increments" was devoted to the case when the random walk is constructed by a fixed sequence of independent random variables which satisfies the classical Lindeberg condition. Now we consider a more general situation when we have a triangular array of independent random variables. Our main assumption is that the entries of every row are uniformly bounded by a constant, which tend…
It is known that the normalized maxima of a sequence of independent and identically distributed bivariate normal random vectors with correlation coefficient $ρ\in (-1,1)$ is asymptotically independent, which may seriously underestimate extreme probabilities in practice. By letting $ρ$ depend on the sample size and go to one with certain rate, Hüsler and Reiss (1989) showed that the normalized maxima can become asymptotically dependent. In this paper, we extend such a study to a triangular array of multivariate Gaussian sequence, which further generalizes the results in Hsing, Hüsler and Reiss …
Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when X 1 has a semiexponential distribution (see, [16, 17]). In the same setting, the authors of [4] derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the t…
We obtain convergence rates (in the Levi-Prokhorove metric) in the functional central limit theorem (CLT) for partial sums $S_n=\sum_{j=1}^{n}ξ_{j,n}$ of triangular arrays $\{ξ_{1,n},ξ_{2,n},...,ξ_{n,n}\}$ satisfying some mixing and moment conditions (which are not necessarily uniform in $n$). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations p…
We study the extreme point process associated to the off-diagonal components in the matrix representation of the Gaussian $β$-Ensemble and prove its convergence to Poisson point process as $n\to +\infty$ when the inverse temperature $β$ scales with $n$ and tends to $0$. We consider two main high temperature regimes: $\displaystyle{β\ll \frac{1}{n}}$ and $\displaystyle{nβ= 2γ\geq 0}$. The normalizing sequences are explicitly given in each cases. As a consequence, we estimate the first order asymptotic of the largest eigenvalue of the Gaussian $β$-Ensemble.
We extend the methods and results of [arXiv 1603.04896] to the setting of multinomial distributions satisfying certain properties. These include all the multinomial distributions arising from the direct proof of the Central Limit Theorem given in [arXiv: 1507.00357], which, by results of that paper, constitutes essentially full generality for the situations in which the Central Limit Theorem holds.
Let $\{ (ξ_{ni}, η_{ni}), 1\leq i \leq n, n\geq 1 \}$ be a triangular array of independent bivariate elliptical random vectors with the same distribution function as $(S_{1}, ρ_{n}S_{1}+\sqrt{1-ρ_{n}^2}S_{2})$, $ρ_{n}\in (0,1)$, where $(S_{1},S_{2})$ is a bivariate spherical random vector. For the distribution function of radius $\sqrt{S_{1}^2+S_{2}^2}$ belonging to the max-domain of attraction of the Weibull distribution, Hashorva (2006) derived the limiting distribution of maximum of this triangular array if convergence rate of $ρ_{n}$ to $1$ is given. In this paper, under the refinement of …
In this paper we show that the componentwise maxima ofweakly dependent bivariate stationary Gaussian triangular arrays converge in distribution after normalisation to Hüsler-Reiss distribution. Under a strong dependence assumption, we prove that the limit distribution of the maxima is a mixture of a bivariate Gaussian distribution and Hüsler-Reiss distribution. Another finding of our paper is that the componentwise maxima and componentwise minima remain asymptotically independent even in the settings of Hüsler and Reiss (1989) allowing further for weak dependence. Further we derive an almost s…
The max-stable Hüsler-Reiss distribution which arises as the limit distribution of maxima of bivariate Gaussian triangular arrays has been shown to be useful in various extreme value models. For such triangular arrays, this paper establishes higher-order asymptotic expansions of the joint distribution of maxima under refined Hüsler-Reiss conditions. In particular, the rate of convergence of normalized maxima to the Hüsler-Reiss distribution is explicitly calculated.
This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the…
We consider a notion of uniform thinning for a finite sequence of random variables $(X_1,...,X_n)$ obtained by removing one random variable, uniformly at random. If a triangular array of random variables $(X_{n,k} : n \in \mathbb{N}_+, 1 \le k \le n)$ satisfies that the law of $(X_{n,1},...,X_{n,n})$ is obtained by uniformly thinning $(X_{n+1,1},...,X_{n+1,n+1})$, then we call the array thinning-invariant. We give a representation for the Choquet simplex of all thinning-invariant triangular arrays of random variables, when all random variables take values in a compact metric space (with Borel …
For 0 < x < 1, take the binary expansion with infinitely many 0's, replace each 0 with -1, this gives the polarized binary expansion of x. Let R_i(x) be the ith "polarized bit" and let S_n(x) be the sum of the first n R_i(x). {S_n} is the Z-valued random walk on (0,1). Normalize, by dividing each S_n by the square root of n: the resulting sequence converges weakly to the standard normal distribution on (0,1). The quantiles of S_n are random variables on (0,1), denoted S*_n, which are equal in distribution to the S_n, Skorokhod showed that the sequence of normalized quantiles converges almost s…
We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically distributed rows. The case of unitary variables is considered as well.
The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,k}, \, 1 \leqslant k \leqslant n, \, n \geqslant 1 \}$ of row-wise extended negatively dependent random variables weakly mean dominated by a random variable $X \in \mathscr{L}_{1}$ and sequences $\{b_{n} \}$ of positive constants, conditions are given to ensure $\sum_{k=1}^{n} \left(X_{n,k} - \mathbb{E} \, X_{n,k} \right)/b_{n} \overset{\textnormal{a.s.}}{\longrightarrow} 0$. Our statements also allow us t…
It is proved that infinitesimal triangular arrays obtained from normalized partial sums of strongly mixing (but not necessarily stationary) random sequences, can produce as lilmits only selfdecomposable distributions.
We give a streamlined proof of the limit theorems for the free additive convolution of infinitesimal triangular arrays of probability measures on the real line. The result was first proved by Chistyakov and Götze using analytic subordination.
