On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Positive dependence properties of elliptically symmetric distributions

Let X₁, …, X_p have p.d.f. g(x₁² + … + x_p²). It is shown that (a) X₁, …, X_p are positively lower orthant dependent or positively upper orthant dependent if, and only if, X₁,…, X_p are i.i.d. N(0, σ²); and (b) the p.d.f. of |X₁|,…, |X_p| is TP₂ in pairs if, and only if, In g(u) is convex. Let X₁, X₂ have p.d.f. $\text{f(x}{}_1\text{, x}{}_2\text{) =}\text{|Σ|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{g((x}{}_1\text{, x}{}_2\text{)}\text{Σ}{}^{\mathrm{?1}}\text{(x}{}_1\text{, x}{}_2\text{)′)}$. Necessary and sufficient conditions are given for f(x₁, x₂) to be TP₂ for fixed correlation ?. It is shown that if f is TP₂ for all ? >0. then (X₁, X₂)′ ～ N(0, Σ). Related positive dependence results and applications are also considered.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

On norm form equations

In his paper [Ann. of Math.96 (1972)] Schmidt applied his results on the n-dimensional Roth theorem to study the n-dimensional Thue equation f(x₁,…, x_n) = c, and he proved, when f is of norm type, a necessary and sufficient criterion for the Thue equation to have only finitely many solutions. Here we shall study equations f(x₁,…, x_n) = p₁^z1 … p_t^zt and prove the p-adic analog of Schmidt's theorems.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Pointwise consistency of the hermite series density estimate

The Hermite series estimate of a density f?L_p, p > 1, convergessin the mean square to f (x) for almost all x? |R, ifN (n) → ∞ and N (n) / n² → ) as n → ∞, where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is O(n^?(2m?1)/2m). Results for complete convergence are also given.     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Canonical row-column-exchangeable arrays

Consider a standard row-column-exchangeable array X = (X_ij : i,j ≥ 1), i.e., X_ij = f(a, ξ_i, η_j, λ_ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′₁, ξ′₂,…, η′₁, η′₂,…, are conditionally independent given ∩_{n ≥ 1}σ(X′_ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.