Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /E[Y _n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y _n ]. These results extend earlier work by the author.     

Rare numbers

Suppose thatX ₁,X ₂,... is a sequence of iid random variables taking values inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofn ^–(1–) Y _n for [0, 1].     

On the Convergence of Sampling-Based Decomposition Algorithms for Multistage Stochastic Programs

The paper presents a convergence proof for a broad class of sampling algorithms for multistage stochastic linear programs in which the uncertain parameters occur only in the constraint right-hand sides. This class includes SDDP, AND, ReSa, and CUPPS. We show that, under some independence assumptions on the sampling procedure, the algorithms converge with probability 1.The first author acknowledges support by the Swiss National Science Foundation. The second author acknowledges support by NZPGST Grant UOAX0203. The authors are grateful to the anonymous referees for comments improving the exposition of this paper.     

混合序列加权和的强收敛性

本文给出混合序列加权和的强收敛性的一些充分条件，这些结论推广和改进了文［１］定理３，文［２］定理３；文［３］定理４．１５以及文［４］定理４．     

Bandwidth Selection in Density Estimation with Truncated and Censored Data

A plug-in type bandwidth selector is presented for density estimation with truncated and censored data. It is based on a representation of the MISE function obtained in the paper. Rate of convergence and limit distribution are derived for this selector. A bootstrap method is introduced to estimate the MISE whose minimizer is an alternative bandwidth selector. A simulation study was carried out to assess the behavior with small samples. This methodology is applied to a real-data problem consisting of reporting delay of AIDS cases. The almost sure representation of the product-limit estimator is a key tool in our proofs.     

鞅差序列加权和的几乎处处收敛(英文)

本文研究了一类鞅差序列加权和的收敛性的问题.利用一些基本不等式和截尾技术,获得了加权和的几乎处处收敛性,推广了关于独立同分布的随机变量序列的相关结果.     

ρ-混合序列部分和乘积的几乎处处极限定理

设{X_n,n≥1}是一严平稳的ρ-混合的正的随机变量序列,且EX_1=μ＞0, Var(X_1)=σ~2,记S_n=Σ_(i=1)~n X_i和γ=σ/μ,在较弱的条件下,证明了对任意的x,,其中σ_1~2=1+2/(σ~2)∑_(j=2)~∞Cov(X_1,X_j),F(·)是随机变量e~(2~(1/2)N)的分布函数,N是标准正态随机变量,我们的结果推广了i．i．d时的情形．     

On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

任意随机变量序列级数的强收敛性

利用鞅差序列级数的收敛定理和条件三级数定理研究了任意随机变量序列级数的强收敛性,推广了某些经典的鞅差序列和独立随机变量序列及两两NQD序列的强极限定理.     

高斯序列的最大值的几乎必然极限

本文研究了高斯序列{Xn}最大值的几乎必然极限。利用正态比较引理和对数平均,在有关协方差的某些条件下,得到了最大值的一个几乎必然极限定理.     

Random series in using unconditional basic sequences and stable sequences: A result on almost sure almost everywhere convergence

Here we study the almost sure almost everywhere convergence of random series of the form in the Lebesgue spaces , where the 's are centered random variables, and the 's constitute an unconditional basic sequence or an stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.

     

Rates of strong uniform consistency for multivariate kernel density estimators

Let f_n denote the usual kernel density estimator in several dimensions. It is shown that if {a_n} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K₁) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on R^d, then for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to . Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities.     

On the global asymptotic stability and oscillation of solutions in a stochastic business cycle model

We discuss a second order nonlinear stochastic difference equation which is constructed of a business cycle model with organized labor considered. A global asymptotic mean square stability criterion is obtained by Lyapunov function method. We also prove a theorem on the almost sure oscillation of the solutions for the difference equation with state-independent stochastic perturbations.     

A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

In this paper we consider the TJW product-limit estimatorF_n(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorF_n(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofF_n(x)−F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.     

Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

随机Volterra积分方程相容解的稳定性

本文考虑了随机Volterra积分方程相容解的稳定性.应用Lyapunov第二方法,并以推广的Ito公式为工具,给出了随机Volterra积分方程相容解的几乎确定指数稳定和矩指数稳定的充分性原则.     

相依序列加权和的收敛性

本文在相依序列下考虑加权和的ａ．ｓ．收敛性和完全收敛性．所得结论推广并改进了［１］、［２］中有关结论．     

一类随机变量序列部分和的强大数定律

设{Xn,n≥1}是随机变量序列.文[4]在二阶矩限制下,获得了任意随机变量序列的Hajek-Renyi型不等式,并给出了随机变量序列的强大数定律.本文利用胡舒合等获得的强大数定律,给出了随机变量序列的一些几乎必然收敛性,并给出了结果在PA,NA和两两NQD序列场合下的应用.     

NA随机变量三角阵列的钟开莱形式的大数定理（英文）

设{Xnm;n≥1,m≥1}每行都是一个NA随机变量列.给出了在各种矩条件随机变量的钟开莱—太勒形式的强大数定理.     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.