Gaussian Central Limit Theorem Research Articles

Gaussian approximation for rooted edges in a random minimal directed spanning tree

AbstractWe study the total ‐powered length of the rooted edges in a random minimal directed spanning tree — first introduced in Bhatt and Roy (2004) — on a Poisson process with intensity on the unit cube for . While a Dickman limit was proved in Penrose and Wade (2004) in the case of , in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when , with a rate of convergence of the order . In this article, we extend these results and prove a central limit theorem in any dimension for any . Moreover, making use of recent results in Stein's method for region‐stabilizing functionals, we provide presumably optimal non‐asymptotic bounds of the order on the Wasserstein and the Kolmogorov distances between the distribution of the total ‐powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.

Universality in the fluctuation of eigenvalues of random circulant matrices

We show that the linear statistics of eigenvalues of random circulant matrices obey the Gaussian central limit theorem for a large class of input sequences.

Energy landscape for large average submatrix detection problems in Gaussian random matrices

The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic analysis of large average submatrices of an $$n \times n$$ Gaussian random matrix. The first part of the paper addresses global maxima. For fixed k we identify the average and the joint distribution of the $$k \times k$$ submatrix having largest average value. As a dual result, we establish that the size of the largest square sub-matrix with average bigger than a fixed positive constant is, with high probability, equal to one of two consecutive integers that depend on the threshold and the matrix dimension n. The second part of the paper addresses local maxima. Specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices. For fixed k, we identify the limiting average value and joint distribution of a $$k \times k$$ submatrix conditioned to be a local maxima. In order to understand the density of such local optima and explain the quick convergence of such iterative procedures, we analyze the number $$L_n(k)$$ of local maxima, beginning with exact asymptotic expressions for the mean and fluctuation behavior of $$L_n(k)$$ . For fixed k, the mean of $$L_{n}(k)$$ is $$\Theta (n^{k}/(\log {n})^{(k-1)/2})$$ while the standard deviation is $$\Theta (n^{2k^2/(k+1)}/(\log {n})^{k^2/(k+1)})$$ . Our principal result is a Gaussian central limit theorem for $$L_n(k)$$ that is based on a new variant of Stein’s method.

On martingale tail sums in affine two-color urn models with multiple drawings

AbstractIn two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Pólya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling

Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are two risk measures which are widely used in the practice of risk management. This paper deals with the problem of estimating both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins-Monro (RM) procedure based on Rockafellar-Uryasev’s identity for the CVaR. Convergence rate of this algorithm to its target satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive and adaptive importance sampling (IS) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which has been investigated by many authors, follows a new approach introduced in [27]. Finally, to speed up the initialization phase of the IS algorithm, we replace the original confidence level of the VaR by a slowly moving risk level. We prove that the weak convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and its efficiency is illustrated on several typical energy portfolios.

Distribution of Hecke eigenvalues

We give two results concerning the distribution of Hecke eigenvalues of SL(2, Z). The first result asserts that on certain average the Sato-Tate conjecture holds. The second result deals with the Gaussian central limit theorem for Hecke eigenvalues.

A gaussian central limit theorem for trimmed products on simply connected step 2-Nilpotent Lie groups

A limit theorem due to J. Kuelbs and M. Ledoux, valid for dilatation-stable laws on type 2-Banach spaces, is carried over to stable laws on simply connected step 2-nilpotent Lie groupsG. We show that for products of i.i.d. random variables in the ℐ of attraction of a nondegenerate ℐ semigroup onG, where ℐ is a one-parameter automorphism group acting contracting onG, a certain “intermediate” trimming procedure, together with a suitable norming, always yields a nondegenerate centered Gaussian limit.

Asymptotic Normality of Trimmed Sums of $\Phi$-Mixing Random Variables

A Gaussian central limit theorem for trimmed sums of $\Phi$-mixing Hilbert-space-valued random variables is obtained, and implications regarding Ibragimov's conjecture are examined.

Some Aspects of the Random walk Model of Stock Market Prices: Comment

REGRETTABLY, Mr. C. W. J. Granger's argument (see Section 3 of the preceding paper) does not suffice to save the central limit theorem with a Gaussian limit. It only shifts the point at which a non-Gaussian limit theorem is required. To represent price data correctly, the instants of transaction must indeed be such that Granger's function N7(t remains a widely scattered random variable even when t is large. To account for my observation that Z(t+ T)-Z(t) has infinite variance, NMt) must even have an infinite expectation. But NVt)/E[N(t)] rapidly tends to one whenever the Gaussian central limit theorem applies to the instants of transaction. Thus, those instants cannot satisfy the Gaussian central limit theorem. A process for which N(t) remains widely scattered even when t is large, studied by Howard M. Taylor and myself [6], is no less strange than the infinite-variance random processes. Economists interested in applying the Gaussian central limit theorem to dependent variables, may ponder the following, still timely, quotation from U. Grenander and M. Rosenblatt [1, (181)]: the experimentalist would argue that in most physically realizable situations where a stationary process has been observed during a time interval long compared to time lags for which correlation is appreciable, the average of the sample would be asymptotically normally distributed. ... Unfortunately none of the extensions of the central limit theorem to dependent variables seems to answer this problem in terms well adapted for practical interpretation. I may add that, if the interdependence between addends is sufficiently strong (but not uinrealistically strong), their weighted sum will not converge to a Gaussian limit. See for example my paper [5], where I describe a process with a Gaussian marginal distribution whose long-term average is not Gaussian but stable Paretian. Such was shown, in [2], to be the case for prices.'