A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Bounds for Non-Gaussian Approximations of U-Statistics

Let X, ,X ₁,...,X _n be i.i.d. random variables taking values in a measurable space ( ). Consider U-statistics of degree two

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if      

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums , of independent random variables. are nonnegative integer-valued random variables independent of r.v. (X _i )i N with distribution functions F _i. We assume that the average of right tails of distribution functions F _i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.     

Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.     

Chevet's theorem for stable processes II

LetB be a separable Banach space and let {:||1} denote the unit ball ofB ^*. LetX be a symmetricp-stableB-valued random variable and let {X _j } _j=1 ⁿ be i.i.d. copies ofX. LetB ₁ be a finite-dimensional Banach space with a symmetric unconditional basis {y _j } _j=1 ⁿ . An upper bound is obtained for that improves the one given by Giné, Marcus and Zinn [J. Functional Anal. 63, 47–73 (1985)].     

Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

On the basis of the exact solution of the linear Dirichlet problem , we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u _tt – u _xx = f(x, t, u, u _t).     

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

On the Hausdorff Dimension of Some Random Cantor Sets

We consider Cantor sets C _X,p constructed by means of a sequence {X _k ^–p } _k=1 of random variables. We find sufficient conditions for the inequality to hold.     

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X _n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,

A priori estimates of strong solutions of semilinear parabolic equations

We study an initial boundary value problem for the semilinear parabolic equation

An extremal problem for empirical measures under dependent Gaussian observations

One describes a class of metricsp in the space of probability distributions on the line, for which the minimum of the mean value of the random variablep(F _x ^* , F _y ^* ), where X, Y are independent random variables, distributed according to the Gauss law , is attained at =I.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 164–166, 1985.The author is grateful to B. S. Tsirel'son for stimulating discussions.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

A Weighted Goodness-of-Fit Test for GARCH(1,1) Specification

Suppose that (j) is the lag-j autocorrelation of the squared residuals computed from a realization of length n under the assumption that the observations follow a GARCH(1,1) model. We study the asymptotic distribution of the statistics of the form , where the _j are nonnegative summable weights and the matrix , can be estimated from the data. We show that, under weak assumptions on model errors, the statistic Q _n converges in distribution to , where the N _i are iid standard normal. We discuss choices of the weights _j for which the distribution of Q is tabulated. Our results lead to and provide a rigorous justification for Portmanteau goodness-of-fit tests for GARCH(1,1) specification.     

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Let μ be the n-dimensional Marcinkiewicz integral and μb the multilinear commutator of μ. In this paper, the following weighted inequalities are proved for ω ∈ A∞ and 0 〈 p 〈 ∞,
｜｜μ（f）｜｜LP（ω）≤C｜Mf｜LP（ω） and ｜｜μb（f）｜｜LP（ω）≤C｜｜ML（log L）^1/r f｜｜LP（ω）.
The weighted weak L（log L）^1/r -type estimate is also established when p=1 and ω∈A1.     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.