Asymptotic Independence of Distant Partial Sums of Linear Processes

We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) for τ ∈ [0, 1], where and are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X _t , t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We consider the cases where (X _t ) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency of certain increment-type statistics in moving-average observations. This work supported by the joint Lithuania-French research program Gilibert. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005.     

Description of sequences of zeros of one class of functions analytic in a half-plane

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition |ƒ(z)| ≤ 0(1) exp (σ| z |η(| z |)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1169–1176, September, 1998.     

Limit theorems for number of diffusion processes, which did not absorb by boundaries

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞. This research was partially supported by the Ministry of Education and Science of Ukraine, project 01.07/103 and University of Salerno, Italy.     

Impact of Bursty Traffic on Queues

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/∞ process. The M/G/∞ process can be seen as a process resulting from the superposition of infinitely many ‘sessions’: sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t, t + 1) is then defined as the number of active sessions at time t (t = 0,1, ...) or, equivalently, as the number of busy servers at time t in an M/G/∞ queue, thereby explaining the terminology. The M/G/∞ process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/∞ process. Second, this process arises naturally in teletraffic as the limiting case for the aggregation of on/off sources [27]. Third, it has been shown to be a good model for various types of network traffic, including telnet/ftp connections [37] and variable-bit-rate (VBR) video traffic [24]. Last but not least, it is amenable to queueing analysis due to its very strong structural properties. In this paper, we compute an asymptotic lower bound for the tail distribution of the queue length. This bound suggests that the queueing delays will dramatically increase as the burstiness of the M/G/∞ input process increases. More specifically, if the tail of G is heavy, implying a bursty input process, then the tail of the queue length will also be heavy. This result is in sharp contrast with the exponential decay rate of the tail distribution of the queue length in presence of ‘non-bursty’ traffic (e.g. Poisson-like traffic). This revised version was published online in August 2006 with corrections to the Cover Date.     

Qualitative investigation of a singular Cauchy problem for a functional differential equation

We consider the singular Cauchy problem

Some first passage time problems for the shortest queue model

We consider the symmetric shortest queue (SQ) problem. Here we have a Poisson arrival stream of rate λ feeding two parallel queues, each having an exponential server that works at rate μ. An arrival joins the shorter of the two queues; if both are of equal length the arrival joins either with probability 1/2. We consider the first passage time until one of the queues reaches the value m ₀, and also the time until both reach this level. We give explicit expressions for the first two first passage moments, conditioned on the initial queue lengths, and also the full first passage distribution. We also give some asymptotic results for m ₀→∞ and various values of ρ=λ/μ. H. Yao work was partially supported by PSC-CUNY Research Award 68751-0037. C. Knessl work was supported in part by NSF grants DMS 02-02815 and DMS 05-03745.     

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues

A fluid queue with ON periods arriving according to a Poisson process and having a long-tailed distribution has long range dependence. As a result, its performance deteriorates. The extent of this performance deterioration depends on a quantity determined by the average values of the system parameters. In the case when the the performance deterioration is the most extreme, we quantify it by studying the time until the amount of work in the system causes an overflow of a large buffer. This turns out to be strongly related to the tail behavior of the increase in the buffer content during a busy period of the M/G/∞ queue feeding the buffer. A large deviation approach provides a powerful method of studying such tail behavior. This revised version was published online in June 2006 with corrections to the Cover Date.     

The limiting distribution of the trace of a random plane partition

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ _n = τ _n (ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ _n − c ₀ n ^2/3)/c ₁ n ^1/3 log^1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c ₀ = ζ(2)/ [2ζ(3)]^2/3, c ₁ = √(1/3/) [2ζ(3)]^1/3 and ζ(s) = Σ _j=1^∞ j ^−s . Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005.     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

Parametric estimation for the standard and geometric telegraph process observed at discrete times

The telegraph process X(t), t ≥ 0, (Goldstein, Q J Mech Appl Math 4:129–156, 1951) and the geometric telegraph process with μ a known real constant and σ > 0 a parameter are supposed to be observed at n + 1 equidistant time points t _i  = iΔ _n ,i = 0,1,..., n. For both models λ, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also σ > 0 has to be estimated. We propose different estimators of the parameters and we investigate their performance under the asymptotics, i.e. Δ _n → 0, nΔ _n =  T < ∞ as n → ∞, with T > 0 fixed. The process X(t) in non markovian, non stationary and not ergodic thus we build a contrast function to derive an estimator. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size n.      

Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium which is an autonomous classical super-Brownian motion. We characterize both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K ^η and mass by K ^−η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.      

Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献   

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Relative entropy and mixing properties of infinite dimensional diffusions

Summary. Let η be a diffusion process taking values on the infinite dimensional space T ^Z , where T is the circle, and with components satisfying the equations dη _i =σ _i (η) dW _i +b _i (η) dt for some coefficients σ _i and b _i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t _n →∞ such that lim _{n →∞}μS _tn =ν exists, where S _t is the semi-group of the process. We prove that if σ _i and b _i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf _{i ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

Asymptotic behaviour of disconnection and non-intersection exponents

Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number of considered paths tends to infinity. In particular, if η _n (respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim _{n →∞} η _n /n = 1 2 and that for a > 0 and b > 0,lim _{n →∞} ξ ([na],[nb])/n = (√ a + √ b) ² /2. Received: 28 February 1996 / In revised form: 3 September 1996     

A Note on Transient Gaussian Fluid Models

In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup _0tT Q(t)>x) as x.     

Anti-integrability for the Logistic Maps

The embedding of the Bernoulli shift into the logistic map x→μx(1- x) forμ> 4 is reinterpreted by the theory of anti-integrability: it is inherited from the anti-integrable limitμ→∞.