The maxima and sums of multivariate non-stationary Gaussian sequences

Let {Xk1, ···, Xkp, k ≥ 1} be a p-dimensional standard(zero-means, unit-variances)non-stationary Gaussian vector sequenceIn this work, the joint limit distribution of the maxima of {Xk1, ···, Xkp, k ≥ 1}, the incomplete maxima of those sequences subject to random failure and the partial sums of those sequences are obtained.     

一类变时滞差分方程的振动性

Consider the nonautonomous delay logistic equation △yn=pnyn(1-n≥0 (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim (n-ln)=∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of (1) to oscillate about k which contains the corresponding result in [2] when I=1.     

OSCILLATION OF CERTAIN nTH ORDER DIFFERENCE EQUATION

1 IntroductionIn this paper, we study the nib order difference equation'L.Xk qkf(X,,) = 0, k E N(0) = {0, 1, 2,' .} (E)where A is the forward difference operator, i.e. axs = xk 1 -- xkj Loxk = xk?L.xk = ac(L.--lxk), in = 1, 2,'', n; {alk},'' I {a.k} are real sequences,ark H I for k E N(0).We set some conditions as followsco(I) afk > 0 for k 2 ho 2 0, Zafk = co, i = 1,2,'' 3n -- 1;(II) {gb} is a real sequence and gb 6 Z = {'' 3 --2, --1, 0, 1, 2,' .} fork E N(0), age 2 0, gb 5 k…     

OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U6（2）

Let G be a finite group and π(G) = { p 1 , p 2 , ··· , p k } be the set of the primes dividing the order of G. We define its prime graph Γ(G) as follows. The vertex set of this graph is π(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈π e (G). In this case, we write p ～ q. For p ∈π(G), put deg(p) := |{ q ∈π(G) | p ～ q }| , which is called the degree of p. We also define D(G) := (deg(p 1 ), deg(p 2 ), ··· , deg(p k )), where p 1 < p 2 < ··· < p k , which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U 6 (2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.     

LEAST SQUARES TYPE ESTIMATION FOR DISCRETELY OBSERVED NON-ERGODIC GAUSSIAN ORNSTEIN-UHLENBECK PROCESSES

In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as d X_t= θX_tdt + dG_t, t ≥ 0 with an unknown parameter θ 0, where G is a Gaussian process. We assume that the process {X_t, t ≥ 0} is observed at discrete time instants t_1 = ?_n, ···, t_n= n?_n, and we construct two least squares type estimators ■ and ■ for θ on the basis of the discrete observations {X_(t_i), i = 1, ···, n}as n →∞. Then, we provide sufficient conditions, based on properties of G, which ensure that ■ and ■ are strongly consistent and the sequences n?n~(1/2)(■-θ) and n?n~(1/2)(■-θ)are tight. Our approach offers an elementary proof of [11], which studied the case when G is a fractional Brownian motion with Hurst parameter H ∈(1/2, 1). As such, our results extend the recent findings by [11] to the case of general Hurst parameter H ∈(0, 1). We also apply our approach to study subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes.     

On a criterion of universal spectra

For positive integers N1 , N2 , . . . , Nn , let L = Zn + A and Λ = N1Z ×···× NnZ + Γ be two rational periodic sets, where A(1/N1 )Z ×···× (1/Nn)Z and ΓZn are finite sets with (A-A) ∩ Zn = {0} and (Γ-Γ) ∩ (N1Z ×···× NnZ) = {0}. In this note, we shall determine conditions under which the tiling set L has universal spectrum Λ. We first obtain a criterion of universal spectra. This criterion combined with the properties of compatible pair yields many necessary and sufficient conditions for Λ to be a universal spectrum for L. We then show that, under some mild additional conditions, the conjecture of Lagarias and Szabó is true. The results here extend the corresponding results of Lagarias, Szabó and Wang.     

STRONG APPROXIMATION FOR MOVING AVERAGE PROCESSES UNDER DEPENDENCE ASSUMPTIONS

Let {Xt,t ≥ 1} be a moving average process defined by Xt = ∑^∞ k=0 αkξt-k, where {αk,k ≥ 0} is a sequence of real numbers and {ξt,-∞ 〈 t 〈 ∞} is a doubly infinite sequence of strictly stationary dependent random variables. Under the conditions of {αk, k ≥ 0} which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, the strong approximation of {Xt, t ≥ 1} to a Gaussian process is studied. Finally, the results are applied to obtain the strong approximation of a long memory process to a fractional Brownian motion and the laws of the iterated logarithm for moving average processes.     

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

A THEOREM ON THE CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES

1 IntroductionThroughout this paperl {X., n 2 1} is assumed to be a sequence of independent randomvariables. Fn denotes the distribution function of the partial surns S. = Z Xk. As is known,k = 1oothe series Z X. is said to be essentially convergent (a.s.) if there exists a sequence of constantsn = 1oo{b., n 2 1} such that Z (X. -- 6.) a.s. converges. When this happens we writeClearlyFn(x) = G.(x -- B.). (l.2)Our problem stems from an old conjecture in Probabilistic Number Theory sugges…     

DURATION OF NEGATIVE SURPLUS FOR A TWO STATE MARKOV-MODULATED RISK MODEL

We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same way. We derive the Laplace transform for the first passage time to surplus zero from a given negative surplus and for the duration of negative surplus. Closed-form expressions are given in the case of exponential individual claim. Finally, numerical results are provided to show how to estimate the moments of duration of negative surplus.     

Parameter estimation and asymptotic stability in stochastic filtering

In this paper, we study the problem of estimating a Markov chain X$X$ (signal) from its noisy partial information Y$Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{_Xn∣_Yn,…,_Y1}$\mathbb{P}\{X_n\mid Y_n,\dots ,Y_1\}$, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.