一类独立随机场的强定律的收敛率

§ 1　IntroductionDefinition1 .[1 ] A field{ Xi,i∈Nd} is called negatively associated(NA) if for every pair ofdisjoint subsets T1 ,T2 of Nd,Cov(f1 (Xi,i∈ T1 ) ,f2 (Xj,j∈ T2 ) )≤ 0 ,whenever f1 and f2 are coordinatewise increasing.Definition2 .[1 ] A field{ Xi,i∈Nd} is calledρ* -mixing ifρ* (s) =sup{ (ρ(S,T) ;S,T N,dist(S,T)≥ s}→ 0 (s→∞ ) ,whereρ(S,T) =sup{ |E(f -Ef) (g -Eg) |/‖ f -Ef‖2 ‖ g -Eg‖2 ,f∈ L2 (σ(S) ) ,g∈ L2 (σ(T) ) } .Definition 3.[1 ] A field { Xi…     

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

PARAMETER ESTIMATION OF SPATIAL AR MODEL

Consider a stable AR model of two parameter spatial series {X_t, t∈N~2}, i. e. {X_(t)t∈N~2} is homogeneous and satisfies the following difference equationX_t-sum from n=s∈相似文献   

Asymptotic properties of plug-in level set estimators for right censored data

We assume T1,...,Tn are i.i.d.data sampled from distribution function F with density function f and C1,...,Cn are i.i.d.data sampled from distribution function G.Observed data consists of pairs(Xi,δi),i=1,...,n,where Xi=min{Ti,Ci},δi=I(Ti Ci),I(A)denotes the indicator function of the set A.Based on the right censored data{Xi,δi},i=1,...,n,we consider the problem of estimating the level set{f c}of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators.Under some regularity conditions,we establish the asymptotic normality and the exact convergence rate of theλg-measure of the symmetric difference between the level set{f c}and its plug-in estimator{fn c},where f is the density function of F,and fn is a kernel-type density estimator of f.Simulation studies demonstrate that the proposed method is feasible.Illustration with a real data example is also provided.     

Comonotone approximation with interpolation at the ends on an interval

Let a function f ∈ C[-1, 1], changes its monotonisity at the finite collection Y := {y1,… ,ys} of s points yi ∈ (-1, 1). For each n ≥ N(Y), we construct an algebraic polynomial Pn, of degree ≤ n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and |f(x)-Pn(x)|≤c(s)ω2(f,(√1-x2)/n), x∈[-1,1],where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s and ω2 (f, t) is the second modulus of smoothness of f.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

Convergence Rates of Wavelet Estimators in Semiparametric Regression Models Under NA Samples

Consider the following heteroscedastic semiparametric regression model:yi =XTiβ + g(ti) + σiei, 1 ＜ i ≤ n,where {Xi,1 ＜ i ＜ n} are random design points,errors {ei,1 ＜ i ＜ n} are negatively associated (...     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

与微分多项式分担值相关的正规定则(英文)

Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D.     

NA序列重对数律的几个极限定理

设{X_n;n≥1}均值为零、方差有限的NA平稳序列。记S_n=∑_(k=1)~n X_k,M_n=maxk≤n|S_k|,n≥1.假设σ~2=EX_1~2+2∑_(k=2)~∞EX_1X_k>0。本文讨论了:当ε 0时,P{M_n≥εσ(2nloglogn)~(1/2)的一类加权级数的精确渐近性质,以及当ε∞时,P{M_n≤εσ(π~2n/(8loglogn))~(1/2)}的一类加权级数的精确渐近性质。这些性质与重对数律和Chung重对数律的速度有关。     

Laws of the iterated logarithm for iterated Wiener processes

The recent interest in iterated Wiener processes was motivated by apparently quite unrelated studies in probability theory and mathematical statistics. Laws of the iterated logarithm (LIL) were independently obtained by Burdzy⁽²⁾ and Révész⁽¹⁷⁾. In this work, we present a functional version of LIL for a standard iterated Wiener process, in the spirit of functional asymptotic results of an ²-valued Gaussian process given by Deheuvels and Mason⁽⁹⁾ in view of Bahadur-Kiefer-type theorems. Chung's liminf sup LIL is established as well, thus providing further insight into the asymptotic behavior of iterated Wiener processes.     

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

Exact rate of convergence in Strassen's law of the interated logarithm

We investigate the upper limiting behavior of the distance of the normalize trajectories of a Wiener process from Strassen's class. It is shown that the right rate is (log logT)^–2/3, improving previous results by the author and by Goodman and Kuelbs.^(2,3)     

自回归模型的Chover型重对数律

利用稳定分布尾概率性质,研究了在扰动为稳定分布的条件下的二阶非稳定回归模型的强收敛性,得到了积分检验的结果,并由此推出了Chover型重对数律,推广了相关文献的结果.     

A functional law of the iterated logarithm for associated sequences

Recently, a functional central limit theorem and a Berry-Essen Theorem have been demonstrated for classes or associated random variables. Using these results, and similar results for multiplicative sequences, we show a functional law of the iterated logarithm for associated sequences satisfying a rate requirement.     

An asymptotic property of gap series. II

An upper bound estimate in the law of the iterated logarithm for Σf(n _k ω) where n_k+1∫n_k≧ 1 + ck ^-α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n _k} is assumed. As an application, the law of the iterated logarithm is proved when {n _k} is the arrangement in increasing order of the set B(τ)={₁ ⁱ ₁...q_τ ⁱ _τ|i₁,...,i_τ∈N ₀}, where τ≧ 2, N ₀=NU{0}, and q ₁,...,q _τ are integers greater than 1 and relatively prime to each others. This revised version was published online in June 2006 with corrections to the Cover Date.     

An asymptotic property of gap series

Takahashi [15] gave a concrete upper bound estimate of the law of the iterated logarithm for ∑ f(n _k x). We extend this result and prove the best possibility of this bound. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Makarov law of the iterated logarithm

We obtain considerable improvement of Makarov's estimate of the boundary behavior of a general conformal mapping from the unit disk to a simply connected domain in the complex plane. We apply the result to improve Makarov's comparison of harmonic measure with Hausdorff measure on simply connected domains.