Extremes of Integer-Valued Moving Average Models with Regularly Varying Tails

In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X _n = _{i = – i} Z _{n – i} where { Z _n} is an iid sequence of non-negative integer-valued random variables with regularly varying tails, _iZ _n–i denotes binomial thinning. Several models are considered allowing different dependence structures on the thinning operations. For these models we manage to establish results which are similar to the ones obtained for the classic linear moving average: { X _n} > has an extremal index = \max_i{ _i } / ( _i ), where is the index of regular variation, irrespectively of the dependence structure considered. The paper is concluded with a simulation study to illustrate the results.     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

多边形螺环链关于Hosoya指标的极值问题

Hosoya指标是化学图论研究中较为流行和重要的拓扑指标之一.首先,根据线性的、无分支的、饱和的多螺环化合物的简单分子结构图定义了"多边形螺环链";其次,研究了多边形螺环链关于Hosoya指标的极值问题;同时,得到了多边形螺环链关于Hosoya指标的排序.     

A Comparison of Methods for Estimating the Extremal Index

The extremal index, (01), is the key parameter when extending discussions of the limiting behavior of the extreme values from independent and identically distributed sequences to stationary sequences. As measures the limiting dependence of exceedances over a threshold u, as u tends to the upper endpoint of the distribution, it may not always be informative about the extremal dependence at levels of practical interest. Therefore we also consider a threshold-based extremal index, (u). We compare the performance of a range of different estimators for and (u) covering processes with < 1 and = 1. We find that the established methods for estimating actually estimate (u), so perform well only when (u) . For Markov processes, we introduce an estimator which is as good as the established methods when (u) but provides an improvement when (u) < = 1. We illustrate our methods using simulated data and daily rainfall measurements.     

Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times

This paper considers a stable GI∨GI∨1 queue with a regularly varying service time distribution. We derive the tail behaviour of the integral of the queue length process Q(t) over one busy period. We show that the occurrence of a large integral is related to the occurrence of a large maximum of the queueing process over the busy period and we exploit asymptotic results for this variable. We also prove a central limit theorem for ∫₀^t Q(s) ds.AMS subject classification: 60K25, 90B22.     

On the sequence of partial maxima of some random sequences

Let {X_n, n ? 1} be a sequence of identically distributed random variables, Z_n = max {X₁,…, X_n} and {u_n, n ? 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Z_n} with respect to {u_n}, in two cases: (1) The X_n's are independent, (2) {X_n} is stationary Gaussian and satisfies a mixing condition.     

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the moment of a sum of independent symmetric random variables to that of the and moments of the individual variables, are computed in the range . This complements the work of Utev who has done the same for . The qualitative nature of the extreme cases turns out to be different for than for . The method developed yields results in some more general and other related moment inequalities.

     

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

On the Structure of Graphs with Given Odd Girth and Large Minimum Degree

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd?s, and Sós implies that every n‐vertex graph with odd girth and minimum degree bigger than must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases and 3, we show that every n‐vertex graph with odd girth and minimum degree bigger than is homomorphic to the cycle of length . This is best possible in the sense that there are graphs with minimum degree and odd girth that are not homomorphic to the cycle of length . Similar results were obtained by Brandt and Ribe‐Baumann.     

On the estimation of the regression coefficients of a continuous parameter process with stationary residual

The asymptotic expressions of the covariance matrices for both the least square estimates _L α _T and Markov (best linear) estimates are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α ₁, …, α _{m 0})′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ _ν(t), t ⩾ 0, ν = 1, …, m ₀, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f(λ). Under these assumptions, it is proven that

On the Cyclic Homology of an Algebra Associated with a Cyclic Quiver and a Monic Polynomial

In this article, we compute the Hochschild homology group of A = KΓ/(f(X ^s )), where KΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K, and X is the sum of all arrows in KΓ. Moreover, we compute the cyclic homology group of A in the case f(x) = (x ? a) ^m , where a ∈ K, so that we can determine the cyclic homology of A in general when K is an algebraically closed field.     

高维电报方程的对称周期解

讨论n维电报方程的对称周期解．利用Schauder不动点定理，在非共振条件下给出了一个存在性定理．     

二维周期系数线性微分方程的特征指数

给出利用系数直接判定二维周期系数线性微分方程特征指数符号的若干结论．     

On the Auslander-Reiten Quiver with OrientedCyclesof Representation-finite Algebras

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＡｂｅａｎａｒｔｉｎａｌｇｅｂｒａ，ｍｏｄＡｂｅｔｈｅｃａｔｅｇｏｒｙｏｆｆｉｎｉｔｅｌｙｇｅｎｅｒａｔｅｄｒｉｇｈｔＡ－ｍｏｄｕｌｅｓ，ΓＡｂｅｉｔｓＡｕｓｌａｎｄｅｒ－Ｒｅｉｔｅｎｑｕｉｖｅｒ，ｔｈｅｎｔｈｅｒｅｉｓａｎｏｎｚｅｒｏｍａｐｂｅｔｗｅｅｎａｎｙｔｗｏｎｏｎ－ｉｓｏｍｏｒｐｈｉｃｍｏｄｕｌｅｓｉｎｍｏｄＡｉｆａｎｄｏｎｌｙｉｆｔｈｅｒｅｉｓａｐａｔｈｂｅｔｗｅｅｎｔｈｅｔｗｏｖｅｒｔｉｃｅｓｃｏｒｒｅｓｐｏｎｄｉｎｇｔｏｔｈｅｓｅｔｗｏｍｏｄ…     

双周期裂纹场平面弹性焊接的数学问题

本文讨论双周期胞腔中含任意形状裂纹的不同材料的弹性平面焊接(焊线为任意形状的封闭光滑曲线)的第二基本问题.运用Мусхелишвили复变函数方法,对这类弹性平面问题建立起了数学模型,将求解弹性平衡问题化归为寻求复应力函数满足一定边界条件的边值问题,然后构造其解的形式,再将其转化为正则型的奇异积分方程,数学上严格证明其解的存在与唯一.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

On the Existence of Positive Periodic Solutions for Neutral Functional Differential Equation with Multiple Deviating Arguments

By means of an abstract continuation theory for κ-set contraction and continuation theorem of coincidence degree principle, some criteria are established for the existence of positive periodic solutions of following neutral functional differential equation dN/dt=N(t)[α(t)-β（t)N(t)-∑j=1^n bj(t)N(t-σj(t))-∑i=1^mci(t)N‘(t-γi(t)0].