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.     

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .     

On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k}}, \hfill \\ {\left| {{p_j} - \sqrt {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)

     

多线性分数次积分算子交换子的有界性

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…     

Operator-Lipschitz functions in Schatten–von Neumann classes

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S ^α , 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant c _α > 0 such that

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y _s = {y _i } ^s _i=1 of points y _i ∈ (-1, 1),

The t-metric Mahler measures of surds and rational numbers

A. Dubickas and C. Smyth introduced the metric Mahler measure

Lp convergence of Cesàro means on sphere

LetR ⁿ be n-dimensional Euclidean space with n>-3. Demote by Ω _n the unit sphere inR ⁿ. ForfɛL(Ω _n ) we denote by σ _N ^δ its Cesàro means of order σ for spherical harmonic expansions. The special value l = \tfracn - 22\lambda = \tfrac{{n - 2}}{2} of σ is known as the critical one. For 0<σ≤λ, we set p₀ = \tfrac2ld+ lp_0 = \tfrac{{2\lambda }}{{\delta + \lambda }} . This paper proves that

Precise asymptotics of complete moment convergence on moving average

Let {ξi,-∞i∞} be a doubly infinite sequence of identically distributed-mixing random variables with zero means and finite variances,{ai,-∞i∞} be an absolutely summable sequence of real numbers and X k =∑i=-∞+∞ aiξi+k be a moving average process.Under some proper moment conditions,the precise asymptotics are established for     

Proof of a conjecture of V. Nikiforov

Using analytical tools, we prove that for any simple graph G on n vertices and its complement [`(G)]\bar G the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1$\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1 holds, where μ(G) and m( [`(G)] )\mu \left( {\bar G} \right) denote the greatest eigenvalue of adjacency matrix of the graphs G and [`(G)]\bar G respectively.     

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.     

On the rates of the other law of the logarithm

Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).     

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

A modular transformation for a generalized theta function with multiple parameters

We obtain a modular transformation for the theta function

On the asymptotical behavior of the constant in the Berry-Esseen inequality

LetF be the distribution function of a sumS _n ofn independent centered random variables, denote the standard normal distribution function and its density. It follows from our results that