Stationary flows and uniqueness of invariant measures

In this short paper, we consider a quadruple $(\Omega, \AA, \theta, \mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $\theta$ is a measurable bijection from $\Omega$ into itself that preserves the measure $\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained by taking cycles (excursions) of iterates of $\theta$ from $B$. We then derive a relation for $\mu_B$ that involves the forward and backward hitting times of $B$ by the trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. We apply the concept to countable Markov chains and Harris processes

Poisson Hail on a Hot Ground

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACS) of the Euclidean space. These RACS arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACS can be served simultaneously and service is in the First In First Out order: only the hailstones in contact with the ground melt at speed 1, whereas the other ones are queued; a tagged RACS waits until all RACS arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.Comment: 26 page

Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

This paper studies the Shannon regime for the random displacement of stationary point processes. Let each point of some initial stationary point process in $\R^n$ give rise to one daughter point, the location of which is obtained by adding a random vector to the coordinates of the mother point, with all displacement vectors independently and identically distributed for all points. The decoding problem is then the following one: the whole mother point process is known as well as the coordinates of some daughter point; the displacements are only known through their law; can one find the mother of this daughter point? The Shannon regime is that where the dimension $n$ tends to infinity and where the logarithm of the intensity of the point process is proportional to $n$. We show that this problem exhibits a sharp threshold: if the sum of the proportionality factor and of the differential entropy rate of the noise is positive, then the probability of finding the right mother point tends to 0 with $n$ for all point processes and decoding strategies. If this sum is negative, there exist mother point processes, for instance Poisson, and decoding strategies, for instance maximum likelihood, for which the probability of finding the right mother tends to 1 with $n$. We then use large deviations theory to show that in the latter case, if the entropy spectrum of the noise satisfies a large deviation principle, then the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is done for two classes of mother point processes: Poisson and Mat\'ern. The practical interest to information theory comes from the explicit connection that we also establish between this problem and the estimation of error exponents in Shannon's additive noise channel with power constraints on the codewords

Zeros of random tropical polynomials, random polytopes and stick-breaking

For $i = 0, 1, \ldots, n$, let $C_i$ be independent and identically distributed random variables with distribution $F$ with support $(0,\infty)$. The number of zeros of the random tropical polynomials $\mathcal{T}f_n(x) = \min_{i=1,\ldots,n}(C_i + ix)$ is also the number of faces of the lower convex hull of the $n+1$ random points $(i,C_i)$ in $\mathbb{R}^2$. We show that this number, $Z_n$, satisfies a central limit theorem when $F$ has polynomial decay near $0$. Specifically, if $F$ near $0$ behaves like a $gamma(a,1)$ distribution for some $a > 0$, then $Z_n$ has the same asymptotics as the number of renewals on the interval $[0,\log(n)/a]$ of a renewal process with inter-arrival distribution $-\log(Beta(a,2))$. Our proof draws on connections between random partitions, renewal theory and random polytopes. In particular, we obtain generalizations and simple proofs of the central limit theorem for the number of vertices of the convex hull of $n$ uniform random points in a square. Our work leads to many open problems in stochastic tropical geometry, the study of functionals and intersections of random tropical varieties.Comment: 22 pages, 5 figure

Generating Functionals of Random Packing Point Processes: From Hard-Core to Carrier Sensing

In this paper we study the generating functionals of several random packing processes: the classical Mat\'ern hard-core model; its extensions, the $k$-Mat\'ern models and the $\infty$-Mat\'ern model, which is an example of random sequential packing process. We first give a sufficient condition for the $\infty$-Mat\'ern model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spaces). Then the generating functional of the resulting point process is given for each of the three models as the solution of a differential equation. Series representations and bounds on the generating functional of the packing models are also derived. Last but not least, we obtain moment measures and Palm distributions of the considered packing models departing from their generating functionals