Probability Inequalities and Limit Theorems for Generalized L-Statistics

We obtain exponential upper bounds for tails of distributions of generalized L-statistics based on a sample from an exponential distribution. We prove the asymptotic normality of generalized L-statistics based on a sample from the uniform distribution on [0,1] and of L-statistics with decomposed kernels (without any restrictions on the sample distribution type).     

The asymptotic number of acyclic digraphs. I

We obtain an asymptotic formula forA _n,q , the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forA _n,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatA _n,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula. Research supported by NSF under grant MCS-8300414. Research supported by NSERC under grant A4067. Research supported by NSF under grant MCS-8302282. Research supported by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowship Scheme, while this author was at the University of Newcastle, Australia.     

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G_Q. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G_Q-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.     

A functional limit theorem for random graphs with applications to subgraph count statistics

We consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. the proof is based on a martingale convergence theorem. the evolving random graph allows us to study both the random graph model K_{n, p}, by fixing attention to a fixed time, and the model K_{n, N}, by studying it at the random time it contains exactly N edges. in particular, we obtain the asymptotic distribution as n → ∞ of the number of subgraphs isomorphic to a given graph G, both for K_{n, p} (p fixed) and K_{n, N} (N/(ⁿ₂)→ p). the results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers of n (the variance grows slower for K_n, N; the powers of n usually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. in some exceptional cases, the limit distribution is not normal.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

L1 regression estimate and its bootstrap

We study the asymptotic distribution of the L ₁ regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution. This work was supported by J.C. Bose National Fellowship, Government of India     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Lower bounds for widths of classes of convolutions of periodic functions in the metrics ofC andL

We give new sufficient conditions for kernels to belong to the set C _y,2n introduced by Kushpel'. These conditions give the possibility of extending the set of kernels belonging to C _y,2n . On the basis of these results, we obtain lower bounds for the Kolmogorov widths of classes of convolutions with these kernels. We show that these estimates are exact in certain important cases.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1112–1121, August, 1995.     

Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L _k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with that of the supremum distance. It turns out that in the uniform case, the measures of deviation are of greater order and their limit distributions are different.     

Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on with coefficients invariant under translation by can be analyzed through decomposition in terms of versions , , of H with z-periodic boundary conditions acting on where . If the s emigroup S generated by H has a H?lder continuous integral kernel satisfying Gaussian bounds then the semigroups generated by the have kernels with similar properties and extends to a function on which is analytic with respect to the trace norm. The sequence of semigroups obtained by rescaling the coefficients of by converges in trace norm to the semigroup generated by the homogenization of . These convergence properties allow asymptotic analysis of the spectrum of H. Received September 1, 1998; in final form January 14, 1999     

A Generalization of the Mean Size Formula of Wavelet Packets in Lp

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

Reliable estimation via simulation

Leta ands denote the inter arrival times and service times in aGI/GI/1 queue. Let a⁽ⁿ⁾, s⁽ⁿ⁾ be the r.v.s. with distributions as the estimated distributions ofa ands from iid samples ofa ands of sizesn. Letw be a r.v. with the stationary distribution of the waiting times of the queue with input(a,s). We consider the problem of estimatingE[w ], > 0 and via simulations when (a ⁽ⁿ⁾,s⁽ⁿ⁾) are used as input. Conditions for the accuracy of the asymptotic estimate, continuity of the asymptotic variance and uniformity in the rate of convergence to the estimate are obtained. We also obtain rates of convergence for sample moments, the empirical process and the quantile process for the regenerative processes. Robust estimates are also obtained when an outlier contaminated sample ofa ands is provided. In the process we obtain consistency, continuity and asymptotic normality of M-estimators for stationary sequences. Some robustness results for Markov processes are included.     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005     

Karhunen-Loeve expansions for the m-th order detrended Brownian motion

The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized (m-th order) Brownian bridge developed by MacNeill (1978) in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L ₂ norm are given.     

Stability and convergence of the mark and cell finite difference scheme for Darcy‐Stokes‐Brinkman equations on non‐uniform grids

In this paper, we consider the mark and cell (MAC) method for Darcy‐Stokes‐Brinkman equations and analyze the stability and convergence of the method on nonuniform grids. Firstly, to obtain the stability for both velocity and pressure, we establish the discrete inf‐sup condition. Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super‐convergence. Finally, we obtain the second‐order convergence in L2 norm for both velocity and pressure for the MAC scheme, when the perturbation parameter ? is not approaching 0. We also obtain the second‐order convergence for some terms of ∥·∥_? norm of the velocity, and the other terms of ∥·∥_? norm are second‐order convergence on uniform grid. Numerical experiments are carried out to verify the theoretical results.     

The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph

The minimal weight of a spanning tree in a complete graph K_n with independent, uniformly distributed random weights on the edges is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph.     

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L ²(w), we obtain a bound that is quadratic in A ₂ constant \({\left[ w \right]_{{A_2}}}\). We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.     

Moderate and Cramér-type large deviation theorems for M-estimators

The known central limit result for broad classes of M-estimators is refined to moderate and large deviation behaviour. The results are applied in relating the local inaccuracy rate and the asymptotic variance of M-estimators in the location and scale problem.