Sums Of Independent Random Vectors Research Articles

Improved central limit theorem and bootstrap approximations in high dimensions

This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional estimation and testing problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.

On a Multivariate Analog of the Zolotarev Problem

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors

The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on an ad hoc basis. Recently, Gaussian approximation has been used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when $n\gg (\log p)^{7}$, where $p$ is the multiplicity of the inference problem and $n$ is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high dimension. In this paper, we reduce the sample size requirement to $n\gg (\log p)^{5}$ for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov–Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anticoncentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable or strong enough to produce desired results.

Synchronized Lévy queues

AbstractWe consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

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.

Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam Region

We initiate the study of the Stam region, defined as the subset of the positive orthant in $\mathbb{R}^{2^n-1}$ that arises from considering entropy powers of subset sums of $n$ independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of A. R. Barron and the first author. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, showing for instance that its closure is a logarithmically convex cone.

On integro-local CLT for sums of independent random vectors

Исследуется остаточный член с учетом асимптотических разложений в интегро-локальном варианте многомерной центральной предельной теоремы для суммы независимых случайных векторов. При этом предполагается, что распределение этой суммы может быть абсолютно непрерывным и (или) решетчатым по некоторым координатам.

A multidimensional analogue of the Rademacher-Gaussian tail comparison

We prove a dimension-free tail comparison between the Euclidean norms of sums of independent random vectors uniformly distributed in centred Euclidean spheres and properly rescaled standard Gaussian random vectors. 2010 Mathematics Subject Classification. Primary 60E15; Secondary 60G15, 60G50.

Performance analysis of averaging based distributed estimation algorithm with additive quantization model

In this paper, we consider the distributed sensor fusion problem over sensor networks under directed communication links and bandwidth constraint. We investigate the impact of the additive quantization model on the proposed two-stage averaging based algorithm. Existing works on the effect of the additive model show that convergence can be guaranteed only if the quantization error variances form a convergent series. We show that the proposed algorithm achieves the performance of the optimal centralized estimate even if the quantization error variances are not vanishing. This is guaranteed by establishing a law of the iterated logarithm for weighted sums of independent random vectors. Moreover, an explicit bound of the convergence rate of the proposed algorithm is given to quantify its almost sure performance.

Estimates of the convergence rate in the “interval” CLT for sums of independent random vectors

Estimates of the convergence rate in the “interval” CLT for sums of independent random vectors

Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where $n$ is the sample size. However, the corresponding bounds on the Kolmogorov distance were only of the order $O(1/n^{1/4})$. In this paper, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform counterparts, which work better in tail zones of the distribution of the MLE. These results are based in part on previously obtained general optimal-order bounds on the rate of convergence to normality in the multivariate delta method. The crucial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of independent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to $M$-estimators.

A weak Cramér condition and application to Edgeworth expansions

We introduce a new, weak Cramér condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramér condition, to distributions that are purely discrete.

A Local Limit Theorem for sums of independent random vectors

We prove a local limit theorem for sums of independent random vectors satisfying appropriate tightness assumptions. In particular, the local limit theorem holds in dimension 1 if the summands are uniformly bounded.

Rosenthal-Type Inequalities for Martingales in 2-Smooth Banach Spaces

Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.

Beyond the Entropy Power Inequality, via Rearrangements

A lower bound on the R\'enyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements. For the special case of Boltzmann-Shannon entropy, this lower bound is better than that given by the entropy power inequality. Several applications are discussed, including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of L\'evy processes.

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Slepian and Sudakov–Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

On the Asymptotic Behavior of Local Probabilities of a Multidimensional Random Walk Crossing Nonlinear Boundaries

This paper proves the theorem on asymptotic behavior of a joint distribution of excess and the first-passage time for a random walk described by a nonlinear function of sums of independent random vectors. As a corollary there are obtained a local limit theorem and asymptotic independence of excess and the first-passage time.

A new strong invariance principle for sums of independent random vectors

We provide a strong invariance principle for sums of independent, identically distributed random vectors that need not have finite second absolute moments. Various applications are indicated. In particular, we show how one can re-obtain some recent LIL type results from this invariance principle. Bibliography: 16 titles.

On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I.

The present paper, consisting of two parts, is sequential to [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594], [A. A. Borovkov and K. A. Borovkov, Theory Probab. Appl., 46 (2002), pp. 193–213 and 49 (2005), pp. 189–206], and [A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks. I. Slowly Decreasing Distributions of Jumps, Nauka, Moscow (in Russian), to be published] and is devoted to studying the asymptotics of the probability that the sum of the independent random vectors is in a small cube with the vertex at point x in the large zone. The papers [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594] are mostly devoted to the “regular deviations problem (the problem [A] using the terminology of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594]), when the relative (“normalized) $x/n$ (n is the number of terms in the sum) a...