Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1–38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537–566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637–662, 2007) by making the first row palindromic. In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure.     

Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables

In this paper, we establish strong laws for weighted sums of identically distributed negatively associated random variables. Marcinkiewicz-Zygmund’s strong law of large numbers is extended to weighted sums of negatively associated random variables. Furthermore, we investigate various limit properties of Cesàro’s and Riesz’s sums of negatively associated random variables. Some of the results in the i.i.d. setting, such as those in Jajte (Ann. Probab. 31(1), 409–412, 2003), Bai and Cheng (Stat. Probab. Lett. 46, 105–112, 2000), Li et al. (J. Theor. Probab. 8, 49–76, 1995) and Gut (Probab. Theory Relat. Fields 97, 169–178, 1993) are also improved and extended to the negatively associated setting.      

Functional BKR Inequalities,and their Duals,with Applications

The inequality conjectured by van den Berg and Kesten (J. Appl. Probab. 22, 556–569, 1985), and proved by Reimer (Comb. Probab. Comput. 9, 27–32, 2000), states that for A and B events on S, a finite product of finite sets, and P any product measure on S,

Time-dependent properties of symmetric queues

We settle a conjecture of Kella et al. (J. Appl. Probab. 42:223–234, 2005): the distribution of the number of jobs in the system of a symmetric M/G/1 queue at a fixed time is independent of the service discipline if the system starts empty. Our derivations are based on a time-reversal argument for regenerative processes and a connection with a clearing model.     

A Moderate Deviation Principle for <Emphasis Type="Italic">m</Emphasis>-Dependent Random Variables with Unbounded <Emphasis Type="Italic">m</Emphasis>

In this paper, a theorem on the moderate deviation principle for random arrays under m-dependence with unbounded m is established. This partially extends the results of Chen (Stat. Probab. Lett. 35:123–134, 1997). As an application, the moderate deviation principle for the truncation estimator of the variance in the analysis of time series is obtained.      

On Quasi-invariance of Infinite Product Measures

Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard Borel space. A characterization of l ²-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat. 36:1107–1112, 1965) and by Hora (Math. Z. 206:169–192, 1991; J. Theor. Probab. 5:71–100, 1992) to all connected Lie groups.      

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

On the regular variation of ratios of jointly Fréchet random variables

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).     

Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation

The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend recent results of Tang (Electron J Probab 11(4):107–120, 2006) and Liu (Stat Probab Lett 79(9):1290–1298, 2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal risk model in which innovations, modelled as real-valued random variables, are negatively dependent with common consistently-varying-tailed distribution, and their inter-arrival times are also negatively dependent.     

Rescaled Lotka–Volterra Models Converge to Super-Stable Processes

Recently, it has been shown that stochastic spatial Lotka–Volterra models, when suitably rescaled, can converge to a super-Brownian motion. We show that the limit process can be a super-stable process if the kernel of the underlying motion is in the domain of attraction of a stable law. The corresponding results in the Brownian setting were proved by Cox and Perkins (Ann. Probab. 33(3):904–947, 2005; Ann. Appl. Probab. 18(2):747–812, 2008). As applications of the convergence theorems, some new results on the asymptotics of the voter model started from single 1 at the origin are obtained, which improve the results by Bramson and Griffeath (Z. Wahrsch. Verw. Geb. 53:183–196, 1980).     

Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609–626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431–457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115–142, 2004) in the L ² region. We prove the convergence of the law of the environment seen by the particle.     

Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem

Consider a Markov additive chain (V,Z) with a negative horizontal drift on a half-plane. We provide the limiting distribution of Z when V passes a threshold for the first time, as V tends to infinity. Our contribution is to allow the Markovian part of an associated twisted Markov chain to be null recurrent or transient. The positive recurrent case was treated by Kesten [Ann. Probab. 2 (1974), 355–386]. Moreover, a ratio limit will be established for a transition kernel with unbounded jumps.     

Several modifications of DPR estimator of the tail index

In this paper, we continue the investigation of an estimator proposed in [Yu. Davydov, V. Paulauskas, and A. Račkauskas, More on p-stable convex sets in Banach spaces, J. Theor. Probab., 13:39–64, 2000] and [V. Paulauskas, A new estimator for tail index, Acta Appl. Math., 79:55–67, 2003] and considered in [V. Paulauskas and M. Vaičiulis, Once more on comparison of tail index estimators, preprint, 2010]. We propose a class of modifications of the so-called DPR estimator and demonstrate that these modifications can have better asymptotic properties than the original DPR estimator.     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue.     

Optimal buffer size for a stochastic processing network in heavy traffic

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A “control cost” related to the dynamically controlled service rate, a “congestion cost” which depends on the queue length and a “rejection penalty” for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b ^*>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005)). To obtain a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we analyze a family of ordinary differential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b ^*>0. A.P. Weerasinghe’s research supported by US Army Research Office grant W911NF0510032.     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

A Generalized Comparison Theorem for BSDEs and Its?Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).     

Exact simulation of the stationary distribution of the FIFO M/G/c queue: the general case for <Emphasis Type="Italic">ρ</Emphasis><<Emphasis Type="Italic">c</Emphasis>

We present an exact simulation algorithm for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ<c. In Sigman (J. Appl. Probab. 48A:209–216, 2011) an exact simulation algorithm was presented but only under the strong condition that ρ<1 (super stable case). We only assume that the service-time distribution G(x)=P(S≤x), x≥0, with mean 0<E(S)=1/μ<∞, and its corresponding equilibrium distribution $G_{e}(x)=\mu\int_{0}^{x} P(S>y)\,dy$G_{e}(x)=\mu\int_{0}^{x} P(S>y)\,dy are such that samples of them can be simulated. Unlike the methods used in Sigman (J. Appl. Probab. 48A:209–216, 2011) involving coupling from the past, here we use different methods involving discrete-time processes and basic regenerative simulation, in which, as regeneration points, we use return visits to state 0 of a corresponding random assignment (RA) model which serves as a sample-path upper bound.