On Some Inequalities for Poisson Networks

A stationary Poisson hyperplane process in R^d induces a random network of (d-2)-flats, each of which is the intersection of two hyperplanes of the process. It is known that the intensity of the induced (d-2)-flat process divided by the square of the intensity of the original hyperplane process is maximal in the isotropic case. An integral-geometric formula for elliptic spaces is presented, from which the mentioned extremum property and related inequalities for superpositions of stationary Poisson hyperplane processes are derived.     

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on R^d with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ^τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ^τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψ^τ has an exponentially decreasing truncated pair correlation function.     

Busy period,virtual waiting time and number of customers in G<Superscript><Emphasis Type="Italic">δ</Emphasis></Superscript>|M<Superscript><Emphasis Type="Italic">ϰ</Emphasis></Superscript>|1|B system

In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected at its infimum. We use the results obtained to study a G ^δ |M ^ϰ |1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.     

Large deviations and the maximum entropy principle for marked point random fields

Summary We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ^d . The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.Supported by the Deutsche Forschungsgemeinschaft     

On convergence to the poisson law for distributions of the sum of random variables connected into a chain

This paper deals with the triangular array of random variables where every sequence is connected into a stationary Markov chain. For the uniformly strongly mixing chain we find conditions under which the distributions of the sum converge to the Poisson law. A more general case of theL ^p-regular Markov chain is also considered. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 297–307, July–September, 1995. Translated by R. Lapinskas     

Cyclically stationary Brownian local time processes

Summary. Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of It?’s law of Brownian excursions. Received: 28 June 1995     

Intersections of random hypersurfaces and visibility

Summary For a wide class of stationary random hypersurfaces in ^d the notion of the projection body is introduced. It turns out that this convex body, a very special case of which is Matheron's Steiner compact associated with a Poisson process of hyperplanes, contains most of the information concerning certain intersection properties of the random hypersurface, while its polar reciprocal set is closely connected with the behaviour of the random hypersurface in visibility problems. This enables one to give a unified treatment of several intersection and visibility problems for random hypersurfaces. A detailed investigation of the case where the random hypersurface is generated by a Poisson process is given separately.     

Refined Approximations for the Ewens Sampling Formula

The Ewens sampling formula is a family of probability distributions over the space of cycle types of permutations of n objects, indexed by a real parameter θ. In the case θ = 1, where the distribution reduces to that induced by the uniform distribution on all permutations, the joint distributions of the numbers of cycles of lengths less than b = o(n) is extremely well approximated by a product of Poisson distributions, having mean 1/j for cycle length j: the error is super-exponentially small with nb^?1. For θ ≠ 1. the analogous approximation, with means adjusted to θ/j, is good, but with error only linear in n^?1b. In this article, it is shown that, by choosing the means of the Poisson distributions more carefully, an error quadratic in n^?1b can be achieved, and that essentially nothing better is possible.     

Closure of Linear Processes

We consider the sets of moving-average and autoregressive processes and study their closures under the Mallows metric and the total variation convergence on finite dimensional distributions. These closures are unexpectedly large, containing nonergodic processes which are Poisson sums of i.i.d. copies from a stationary process. The presence of these nonergodic Poisson sum processes has immediate implications. In particular, identifiability of the hypothesis of linearity of a process is in question. A discussion of some of these issues for the set of moving-average processes has already been given without proof in Bickel and Bühlmann.⁽²⁾ We establish here the precise mathematical arguments and present some additional extensions: results about the closure of autoregressive processes and natural sub-sets of moving-average and autoregressive processes which are closed.Research supported in part by grants NSA MDA 904-94-H-2020 and NSF DMS 95049555     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

Nonparametric Estimation of Regression Functions in Point Process Models

We prove that the empirical L ²-risk minimizing estimator over some general type of sieve classes is universally, strongly consistent for the regression function in a class of point process models of Poissonian type (random sampling processes). The universal consistency result needs weak assumptions on the underlying distributions and regression functions. It applies in particular to neural net classes and to radial basis function nets. For the estimation of the intensity functions of a Poisson process a similar technique yields consistency of the sieved maximum likelihood estimator for some general sieve classes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Vertex Numbers of Weighted Faces in Poisson Hyperplane Mosaics

In the random mosaic generated by a stationary Poisson hyperplane process in ℝ ^d , we consider the typical k-face weighted by the j-dimensional volume of the j-skeleton (0≤j≤k≤d). We prove sharp lower and upper bounds for the expected number of its vertices.     

On the Number of Lost Customers in Stationary Loss Systems in the Light Traffic Case

For the stationary loss systems M/M/m/K and GI/M/m/K, we study two quantities: the number of lost customers during the time interval (0,t] (the first system only), and the number of lost customers among the first n customers to arrive (both systems). We derive explicit bounds for the total variation distances between the distributions of these quantities and compound Poisson–geometric distributions. The bounds are small in the light traffic case, i.e., when the loss of a customer is a rare event. To prove our results, we show that the studied quantities can be interpreted as accumulated rewards of stationary renewal reward processes, embedded into the queue length process or the process of queue lengths immediately before arrivals of new customers, and apply general results by Erhardsson on compound Poisson approximation for renewal reward processes.     

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ℝ₊ ⁿ with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes. F.J. Piera’s research supported in part by CONICYT, Chile, FONDECYT Project 1070797. R.R. Mazumdar’s research supported in part by NSF, USA, Grant 0087404 through Networking Research Program, and a Discovery Grant from NSERC, Canada.     

Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes

A stationary Poisson cylinder process Π _cyl ^(d,k) is composed of a stationary Poisson process of k-flats in ℝ ^d that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume V _ϱ ^(d,k) of the union set of Π _cyl ^(d,k) that covers ϱW as the scaling factor ϱ becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance of V _ϱ ^(d,k) that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of V _ϱ ^(d,k) under the assumption that the (d − k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated “Lemma on large deviations” of Statulevičius.     

Limit distributions of sums of m-dependent Bernoulli random variables

Summary The set of limit distributions of row sums of a triangular array of Bernoulli random variables which is strictly stationary and m-dependent in each row is characterized. Necessary and sufficient conditions for the convergence of the row sums to a given limit distribution are found. The case of convergence to a Poisson distribution is given special attention.     

Insensitivity of multiclass systems with general dynamic preemptive resume queueing disciplines

We are concerned with the insensitivity of the stationary distributions of the system states inM/G/s/m queues with multiclass customers and with LIFO preemptive resume service disciplines. We introduce general entrance and exit rules into and from waiting positions, respectively, for the behaviour of waiting customers whose service is interrupted. These rules may, roughly speaking, depend on the number of customers in the system. It is shown that the stationary distribution of the system state is insensitive not only with respect to the service time distributions but also with respect to the general entrance and exit rules. As well as the insensitivity of the service scheme, our results are obtained for a special form of state and customer type dependent arrival and service rates. Some further results are concluded related to insensitivity like the formula for the conditional mean sojourn time and the property of transformation of a Poisson input into a Poisson output by the systems.     

Extreme value theory for continuous parameter stationary processes

Summary In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences.The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite mean(x), it is found that the classical criteria for domains of attraction apply when(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for which(x) may or may not be finite).A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr ^th largest local maxima.This work was supported by the Office of Naval Research under Contract N00014-75-C-0809, and in part by the Danish natural Science research Council     

On the Stationary Distribution of the GI X /M Y /1 Queueing System

For the GI ^X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI ^X /M ^Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.     

The M/M/1 queue with inventory, lost sale, and general lead times

We consider an M/M/1 queueing system with inventory under the $(r,Q)$ policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.