On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables

Let {Xni, 1 ≤ n,i ＜∞} be an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 ＜ an ↑∞. The limiting behavior of maximum partial sums 1/an max 1≤k≤n| kΣi=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].     

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {Vn , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞∑ n i=1 Vi /gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.     

Large Deviations for Random Sums on Some Kind of Heavy-tailed Classes in Risk Models

This paper is a further investigation into the large deviations for random sums of heavy-tailed,we extended and improved some results in ref. [1] and [2]. These results can applied to some questions in Insurance and Finance.     

THE DECOMPOSITION OF STATE SPACE FOR MARKOV CHAIN IN RANDOM ENVIRONMENT

This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random walk in random environment, renewal process in random environment and queue process in random environment, and obtains the decompositions of the state spaces of these three special examples.     

Lr Convergence for Arrays of Rowwise Negatively Sup eradditive Dep endent Random Variables

Let {Xnk, k≥1, n≥1} be an array of rowwise negatively superadditive depen-dent random variables and {an, n ≥ 1} be a sequence of positive real numbers such that an ↑ ∞. Under some suitable conditions, Lr convergence of a1n 1max≤j≤n ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables. fl fl fl fl jP k=1 Xnk fl fl flfl is stud-ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.     

THE ANALYTICAL PROPERTIES FOR HOMOGENEOUS RANDOM TRANSITION FUNCTIONS

The concepts of Markov process in random environment and homogeneous random transition functions are introduced. The necessary and sufficient conditions for homogeneous random transition function are given. The main results in this article are the analytical properties, such as continuity, differentiability, random Kolmogorov backward equation and random Kolmogorov forward equation of homogeneous random transition functions.     

Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.     

Precise asymptotics for random matrices and random growth models

The author considers the largest eigenvaiues of random matrices from Gaussian unitary ensemble and Laguerre unitary ensemble, and the rightmost charge in certain random growth models. We obtain some precise asymptotics results, which are in a sense similar to the precise asymptotics for sums of independent random variables in the context of the law of large numbers and complete convergence. Our proofs depend heavily upon the upper and lower tail estimates for random matrices and random growth models. The Tracy-Widom distribution plays a central role as well.     

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.     

NSD随机变量阵列的L_r收敛性（英文）

Let {X_(nk), k ≥ 1, n ≥ 1} be an array of rowwise negatively superadditive dependent random variables and {a_n, n ≥ 1} be a sequence of positive real numbers such that a_n↑∞. Under some suitable conditions,L_r convergence of 1/an max 1≤j≤n |j∑k=1 X_(nk)| is studied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.     

CONVERGENCE ON RANDOMLY TRIMMED SUMS WITH A DEPENDENT SAMPLE

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔｓＬｅｔ｛Ｘｎ，ｎ１｝ｂｅａｓｅｑｕｅｎｃｅｏｆｒａｎｄｏｍｖａｒｉａｂｌｅｓｗｉｔｈａｃｏｍｍｏｎｄｉｓｔｒｉｂｕｔｉｏｎｆｕｎｃｔｉｏｎＦ（ｘ）ａｎｄｌｅｔＸｎ１Ｘｎ２…Ｘｎｎｂｅｔｈｅｏｒ...     

在最少条件下的对数律精确渐近性

设$X_1,X_2,\cdots$为独立同分布随机变量, 记S_n=X_1+\cdots+X_n,\;M_n=\max\limits_{k\le n}|S_k|,\;n\ge1$. 本文在充分必要条件下给出了$M_n$和$S_n$的对数律之精确渐近性.     

A Bijective Answer to a Question of Zvonkin

The purpose of this note is to give a bijective proof of the identity< /FORMULA > < FORMULA > E \left[ \prod\limits_{1\le iFORMULA > < FORMULA >where X₁,..., X_n are independent identically distributed normal random variables with mean 0 and variance 1. The bijection is obtained by combining a bijection of Gessel and a bijection of Ehrenborg with the interpretation that the moments of the normal distribution count the number of matchings.     

A maximal inequality for partial sums of finite exchangeable sequences of random variables

Let be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant such that if , then

for all . This generalizes an inequality of Montgomery-Smith and Lata{\l}a for independent and identically distributed random variables. Our maximal inequality is apparently new even if is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices.

     

ON SMOOTHNESS AND DIFFERENTIABILITY OF FUNCTIONS

Let \(f(x)\) be a bounded real function on [-1,1],we define the modulus of continuity of f as \[\omega (f,\delta ) = \mathop {\sup }\limits_{x,y \in [ - 1,1],\left| {x - y} \right| \le \delta } \left| {f(x) - f(y)} \right|\] and the modulus of smoothness of f as \[{\omega _2}(f,\delta ) = \mathop {\sup }\limits_{x \pm h \in [ - 1,1],\left| h \right| \le \delta } \left| {f(x + h) + f(x - h) - 2f(x)} \right|\] Functions \(f(x)\), continuous on [-1,1] and \({\omega _2}(f,\delta ) = o(\delta )\) ,are called uniformly smooth functions. It is well known that there is a uniformly smooth functions whose derivative exisits on a null-set only. It would is of interest to discuss what condition should be added on the nonnegative function \(\varphi (\delta )\), \(\left( {0 \le \delta \le \frac{1}{2}} \right)\),in order that every bounded function f satisfying\[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative. the main result of this paper are the following two theorems. Theorem 1 let \(\varphi (\delta )\),\(\left( {0 \le \delta \le \frac{1}{2}} \right)\) ,be a nonnegative function, then, in order that every bounded function \(f(x)\) satisfying condition \[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative \(f'(x)\) on [-1,1],it is necessary and sufficient that the following condition hold \[\int_0^{\frac{1}{2}} {\frac{{\tilde \varphi (t)}}{t}} dt < \infty \] where \[\tilde \varphi (\delta ) = {\delta ^2}\mathop {\inf }\limits_{0 \le \eta \le \delta } \left\{ {{\eta ^{ - 2}}\mathop {\inf }\limits_{\eta \le \xi \le 1/2} \varphi (\xi )} \right\}\] Theorm 2 Let \(f(x)\) be a bounded function with \[\int_0^{\frac{1}{2}} {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt < \infty \] then \(f'(x)\) is a continous function and \[{\omega _2}(f',\delta ) = O\left\{ {\int_0^\delta {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt} \right\}\].     

ON THE GENERALIZED POLYNOMIALS OF L. V. KANTOROVITCH AND THEIR ASYMPTOTIC BEHAVIORS

In this article we generahze the polynomials of Kantorovitch \({P_n}(f)\) . Let \({B_n}\) be a sequence of linear operators from C[a,b] into \({H_n}\), if \[f(t) \in L[a,b],F(u) = \int_a^u {f(t)dt} ,{A_n}(f(t),x) = \frac{d}{{dx}}{B_{n + 1}}(F(u),x)\], here \({B_n}\)satisfy\[\begin{array}{l} (a):{B_n}(1,x) \equiv 1,{B_n}(u,x) \equiv x;\(b):for{\kern 1pt} {\kern 1pt} g(u) \in C[a,b]{\kern 1pt} {\kern 1pt} we{\kern 1pt} {\kern 1pt} have{\kern 1pt} {\kern 1pt} {B_n}(g(u),b) = g(b). \end{array}\]. we call such \({A_n}(f)\) generalized polynomials of Kantorovitch (denoted by \({A_n}(f) \in K\) ). Let \[\begin{array}{l} {\varepsilon _n}({W^2};x)\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}} \left| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right|,\{\varepsilon _n}{({W^2}{L^p})_{{L^p}}}\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}{L^p}} {\left\| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right\|_p}. \end{array}\] We have proved the following results: Let An he a sequence of linear continuous operators of type \[C[a,b] \Rightarrow C[a,b],{D_n}(x,z)\mathop = \limits^{def} {A_n}(\left| {t - z} \right|,x) - \left| {x - z} \right| - ({A_n}(t,x) - x)Sgn(x - z),{A_n}(1,x) = 1\] then (1):\({\varepsilon _n}({W^2};x) = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dz\), (2): Moreover, if \({A_n}\) be a sequence of linear positive operators, then for \(\left[ {\begin{array}{*{20}{c}} {a \le x \le b}\{a \le z \le b} \end{array}} \right]\) ,we have \({D_n}(x,z) \ge 0\), and \({\varepsilon _n}({W^2};x) = \frac{1}{2}{A_n}({(t - x)^2},x)\). Let \({A_n}(f) \in K\) be a sequence of linear positive operators,\[{R_n}{(z)_L} = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dx\],then \[{R_n}{(z)_L} = \frac{1}{2}\left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right]\] and \[{\varepsilon _n}{({W^2}L)_L}{\rm{ = }}\frac{1}{2}\left\| {{B_{n + 1}}({u^2},z) - {z^2}} \right\|\]. Let \[{g_n} = \frac{1}{2}\mathop {\max }\limits_{a \le x \le b} {A_n}({(t - x)^2},x),{h_n} = \frac{1}{2}\mathop {\max }\limits_{a \le z \le b} \left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right],\] then \[{\varepsilon _n}{({W^2}{L^p})_{{L^p}}} \le {g_n}^{1 - \frac{1}{p}}{h_n}^{\frac{1}{p}}(1 < p < \infty ).\]     

Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion

Consider the fractional Brownian motion process $B_H(t), t\in [0,T]$, with parameter $H\in (0,1)$. Meyer, Sellan and Taqqu have developed several random wavelet representations for $B_H(t)$, of the form $\sum_{k=0}^\infty U_k(t)\epsilon_k$ where $\epsilon_k$ are Gaussian random variables and where the functions $U_k$ are not random. Based on the results of Kühn and Linde, we say that the approximation $\sum_{k=0}^n U_k(t)\epsilon_k$ of $B_H(t)$ is optimal if $$ \displaystyle \left( E \sup_{t\in [0,T]} \left| \sum_{k=n}^\infty U_k(t) \epsilon_k\right|^2 \right)^{1/2} =O \left( n^{-H} (1+\log n)^{1/2} \right), $$ as $n\rightarrow\infty$. We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.     

On the max-domain of attractions of bivariate elliptical arrays

Let be a triangular array of independent bivariate elliptical random vectors with the same distribution function as where is a bivariate spherical random vector. Under assumptions on the speed of convergence of we show in this paper that the maxima of this triangular array is in the max-domain of attraction of a new max-id. distribution function , provided that has distribution function in the max-domain of attraction of the Weibull distribution function . AMS 2000 Subject Classification Primary—60G70, 60F05     

Boundedness of Solutions for Duffing Equation with Low Regularity in Time

It is shown that all solutions are bounded for Duffing equation x+ x~(2n+1)+2∑i=nPj(t)x~j= 0, provided that for each n + 1 ≤ j ≤ 2 n, P_j ∈ C~y(T~1) with γ 1-1/n and for each j with 0 ≤ j ≤ n, Pj ∈ L(T~1) where T~1= R/Z.     

A Note on Bounds for the Supremum Metric for Discrete Random Variables

For positive or negative integer-valued random variables X and Y with finite second moments the inequality sup \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\sup }\limits_n |\Pr \{ X \le n\} - \Pr \{ Y \le n\} |\, \le \,|EX - EY| + \frac{1}{2}(EX(EX - 1) + (EY(Y - 1)) $\end{document} is established by elementary manipulation, and shown to be tight. Use of generating functions and an inversion formula yields the larger bound with 1/2 replaced by 2/π.