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Statistical evidence is presented to answer the title question using graphical tools from process behavior charting as well as ranking methods based on principal component analysis. These tools provide strong data evidence to answer the question convincingly. [ABSTRACT FROM AUTHOR]

The geometric process is a popular method for the modeling of arrival times with trend. In this study, the statistical inference problem for geometric process is considered when the distribution of first arrival time is two-parameter Lindley. The parameters a, λ and β are estimated using the maximum likelihood, modified least-squares, modified moments and modified L-moments methods. The estimation performances of the obtained estimators are compared by comprehensive simulations. The simulation results show that maximum likelihood estimators outperform the other estimators. Finally, on three real-life datasets, we present data analyses showing the modeling performance of geometric process against renewal process. [ABSTRACT FROM AUTHOR]

In this paper, motivated by the risk process, we introduce and study a new point process called geometric Pólya-Aeppli process (GPAP), with underlying exponential distribution. We give two equivalent definitions of the process and discuss some of its properties, such as the distribution of the number of GPAP events up to time t, the distribution of the waiting times, etc. The new process is an extension of the well-known Pólya-Aeppli process, as well as the standard geometric process with underlying exponential distribution. [ABSTRACT FROM AUTHOR]

The paper addresses a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. These equations are allowed to live in Euclidean spaces of different dimensions, and the system is Markovian in the sense that the terminal value of the backward equation depends on the terminal value of the solution of the forward one. Under some monotonicity conditions we establish the existence and uniqueness of strong solutions of such equations by using a continuation method. [ABSTRACT FROM AUTHOR]

The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to first-order target count estimates. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson point process, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided. [ABSTRACT FROM AUTHOR]

We investigate local statistics of eigenvalues for random normal matrices, represented as 2D determinantal Coulomb gases, in the case when the eigenvalues are forced to be in the support of the equilibrium measure associated with an external field. For radially symmetric external fields with sufficient growth at infinity, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues. [ABSTRACT FROM AUTHOR]

We present a new approach to study big data in finance (specifically, in limit order books), based on stochastic modelling of price changes associated with high-frequency and algorithmic trading. We introduce a big data in finance, namely, limit order books (LOB), and describes them by Lobster data-academic data for studying LOB. Numerical results, associated with Lobster and other data, are presented, and explanation and justification of our method of studying of big data in finance are considered. We also describe various stochastic models for mid-price changes in the market, and explain how to use these models in practice, highlighting the methodological aspects of using the models. [ABSTRACT FROM AUTHOR]

Matérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process. [ABSTRACT FROM AUTHOR]

Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of K, a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm. [ABSTRACT FROM AUTHOR]

We introduce a family of boundary confinements for Coulomb gas ensembles, and study them in the two-dimensional determinantal case of random normal matrices. The family interpolates between the free boundary and hard edge cases, which have been well studied in various random matrix theories. The confinement can also be relaxed beyond the free boundary to produce ensembles with fuzzier boundaries, i.e., where the particles are more and more likely to be found outside of the boundary. The resulting ensembles are investigated with respect to scaling limits and distribution of the maximum modulus. In particular, we prove existence of a new point field—a limit of scaling limits to the ultraweak point when the droplet ceases to be well defined. [ABSTRACT FROM AUTHOR]

We consider the simultaneous optimization of the reliability and the cost of a ceramic component in a biobjective PDE constrained shape optimization problem. A probabilistic Weibull-type model is used to assess the probability of failure of the component under tensile load, while the cost is assumed to be proportional to the volume of the component. Two different gradient-based optimization methods are suggested and compared at 2D test cases. The numerical implementation is based on a first discretize then optimize strategy and benefits from efficient gradient computations using adjoint equations. The resulting approximations of the Pareto front nicely exhibit the trade-off between reliability and cost and give rise to innovative shapes that compromise between these conflicting objectives. [ABSTRACT FROM AUTHOR]

We consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator. [ABSTRACT FROM AUTHOR]

The PD α (r) distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the PD α distribution introduced by Kingman, to which it reduces when r = 0 . The parameter α ∈ (0 , 1) arises from its construction based on ratios of ordered jumps of an α -stable subordinator, and the parameter r > 0 signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution PD (θ) , occur as limiting cases of PD α (r) , as r → ∞ . As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made. [ABSTRACT FROM AUTHOR]

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO (3) , with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO (3) . The variance of rotation matrices sampled by a certain determinantal point process is estimated, and formulas for the L 2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest. [ABSTRACT FROM AUTHOR]

The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non‐periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an EI equal to 0 with that of an EI larger than 0. Using bi‐dimensional point processes, we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the cluster size distribution introduced earlier and prove that ergodicity is enough to establish that the finite versions of the reciprocal of the EI and of the mean of the cluster size distribution do coincide. [ABSTRACT FROM AUTHOR]

For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index α ∈ (0 , 2) and weak dependence conditions. The limiting process consists of an α-stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of R 2 -valued càdlàg functions on [ 0 , 1 ] , with the Skorohod weak M 1 topology. We further show that this topology in general can not be replaced by the stronger (standard) M 1 topology. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We derive expansions for the bias and variance in the scenario that n independent copies of a point process in ℝ d are superposed. When the same bandwidth is used in all d dimensions, we show that an optimal bandwidth exists and is of the order n− 1/(d+ 4) under appropriate smoothness conditions on the true intensity function. [ABSTRACT FROM AUTHOR]

We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes. [ABSTRACT FROM AUTHOR]

We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts. [ABSTRACT FROM AUTHOR]

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes. [ABSTRACT FROM AUTHOR]

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/ $\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem. [ABSTRACT FROM AUTHOR]

We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of the k-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation. [ABSTRACT FROM AUTHOR]

The concept of hyperuniformity has been introduced by Torquato and Stillinger in 2003 as a notion to detect structural behaviour intermediate between crystalline order and amorphous disorder. The present paper studies a generalisation of this concept to the unit sphere. It is shown that several well studied determinantal point processes are hyperuniform. [ABSTRACT FROM AUTHOR]

We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3, 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincaré Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form - 1 2 Δ + V + ξ , where V is a deterministic potential and ξ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise ξ . [ABSTRACT FROM AUTHOR]

We wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod 𝔻 {\mathbb{D}} equipped with the S-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems. [ABSTRACT FROM AUTHOR]

We analyze the time-dependent behavior of various types of infinite-server queueing systems, where, within each system we consider, jobs interact with one another in ways that induce batch departures from the system. One example of such a queue was introduced in the recent paper of Frolkova and Mandjes (Stochastic Models, 2019) in order to model a type of one-sided communication between two users in the Bitcoin network: here we show that a time-dependent version of the distributional Little's law can be used to study the time-dependent behavior of this model, as well as a related model where blocks are communicated to a user at a rate that is allowed to vary with time. We also show that the time-dependent behavior of analogous infinite-server queueing systems with batch arrivals and exponentially distributed services can be analyzed just as thoroughly. [ABSTRACT FROM AUTHOR]

We consider the Voronoi diagram generated by n independent and identically distributed $\mathbb{R}^{d}$ -valued random variables with an arbitrary underlying probability density function f on $\mathbb{R}^{d}$ , and analyze the asymptotic behaviors of certain geometric properties, such as the measure, of the Voronoi cells as n tends to infinity. We adapt the methods used by Devroye et al. (2017) to conduct a study of the asymptotic properties of two types of Voronoi cells: (1) Voronoi cells that have a fixed nucleus; (2) Voronoi cells that contain a fixed point. We show that the geometric properties of both types of cells resemble those in the case when the Voronoi diagram is generated by a homogeneous Poisson point process. Additionally, for the second type of Voronoi cells, we determine the limiting distribution, which is universal in all choices of f, of the re-scaled measure of the cells. [ABSTRACT FROM AUTHOR]

The equilibrium properties of allocation algorithms for networks with a large number of nodes with finite capacity are investigated. Every node receives a flow of requests. When a request arrives at a saturated node, i.e. a node whose capacity is fully utilized, an allocation algorithm may attempt to reallocate the request to a non-saturated node. For the algorithms considered, the reallocation comes at a price: either extra capacity is required in the system, or the processing time of a reallocated request is increased. The paper analyzes the properties of the equilibrium points of the associated asymptotic dynamical system when the number of nodes gets large. At this occasion the classical model of Gibbens, Hunt, and Kelly (1990) in this domain is revisited. The absence of known Lyapunov functions for the corresponding dynamical system significantly complicates the analysis. Several techniques are used. Analytic and scaling methods are used to identify the equilibrium points. We identify the subset of parameters for which the limiting stochastic model of these networks has multiple equilibrium points. Probabilistic approaches are used to prove the stability of some of them. A criterion of exponential stability with the spectral gap of the associated linear operator of equilibrium points is also obtained. [ABSTRACT FROM AUTHOR]

We give a new integral characterization of the Dirichlet process on a general phase space. To do so, we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson–Dirichlet distribution via invariance and independence properties. [ABSTRACT FROM AUTHOR]

We construct marked Gibbs point processes in R d under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks—attached to the locations in R d —belong to a general normed space S . They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented. [ABSTRACT FROM AUTHOR]

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where β N → c o n s t ∈ (0 , ∞) , with N the system size and β the inverse temperature. For the global behavior, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle. This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process. [ABSTRACT FROM AUTHOR]

In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk. [ABSTRACT FROM AUTHOR]

Future wireless networks like 5G will carry an increasingly wide variety of data traffic, with different QoS requirements. In addition to conventional data traffic generated from HTTP, FTP and video streaming applications by mobile broadband users [human-type communication (HTC)], traffic from machine-to-machine (M2M) and Internet-of-Things (IoT) applications [machine-type communication (MTC)] has to be supported by 5G networks. Time-of-day variation in arrival rate of connection-level requests and randomness in length of data sessions in HTC result in randomness in aggregate traffic. In MTC, randomness in traffic arises from random number of devices trying to connect to the base station at any given time. Traffic generated by MTC devices may be either periodic or event-triggered. Nevertheless, it is difficult to model aggregate traffic due to non-stationary nature of traffic generated by each type of service. In this paper, special correlation functions of stochastic point processes called Product Densities (PDs) are used for estimating aggregate traffic under non-stationary arrival rates. For HTC, PDs are defined for estimating time-dependent offered load of connection-level service requests and expected number of ON periods in an interval of time (0 , T) . The aggregate traffic is evaluated for light-tail (exponential) and heavy-tail (hyper exponential) servicing times. For MTC, PDs are defined for estimating the random number of devices connected to the base station at any time. Another QoS parameter of interest in high speed networks is the expected number of service requests/devices delayed beyond a critical value of delay. Bi-variate PD is defined to estimate the number of service requests/devices delayed beyond a given critical threshold. The results from PD model show close agreement with simulation results. The proposed PD technique proves effective in performance analysis under time-dependent traffic conditions, and is versatile for application to several studies in wireless networks including power consumption, interference and handover performance. [ABSTRACT FROM AUTHOR]

We analyze the behavior of a particle moving along a d-dimensional lattice and representing a "doubly stochastic" random walk. Unlike the classical random walk, the lattice is randomly generated upon particle's landing at a node. At any time, the particle is enclosed in a rectangular cylinder R = R a × R b (i.e., with a bounded a-dimensional rectangle Ra at its base) it attempts to escape. Thus, the particle moves from one node to another at random epochs of time and, with some probability, leaves R. The particle may jump arbitrarily far beyond the boundary ∂ R at its passage time. Furthermore, the particle's location is not assumed to be known in real time, but only upon certain random epochs { τ n }. Of a key interest is the location A (t) of the particle at any time, where A (t) is the linear interpolation of particle's positions at times τn's. If τ ρ denotes the first observed escape time (virtual first passage time), where ρ = min { n : A ( τ n) ∈ R C } , we target the joint characteristic function of A (τ ρ − 1) , A (τ ρ) , τ ρ − 1 , τ ρ , and A (t) itself, where t ∈ [ 0 , τ ρ − 1 ] or (τ ρ − 1 , τ ρ ] in tractable forms, thereby attempting to enhance as much as possible the probabilistic data lost due to the crudeness of the observations and to couple A (τ ρ − 1) , A (τ ρ) , τ ρ − 1 , and τ ρ with deterministic time intervals [ 0 , t ]. Among various applications, we discuss and treat antagonistic games of two active and several passive players as well as situations that occur in complex queueing systems. [ABSTRACT FROM AUTHOR]

We introduce a new model for pricing corporate bonds, which is a modification of the classical model of Merton. In this new model, we drop the liquidity assumption of the firm's asset value process, and assume that there is a liquidly tradeable asset in the market whose value is correlated with the firm's asset value, and all portfolios can be constructed using solely this asset and the money market account. We formulate the market price of the corporate bond as the product of the price of an optimal replicating portfolio and exp ⁡ (− κ × replication error) , where κ is a non-negative constant. The interpretation is that the representative investor uses the price of the optimal replicating portfolio as a benchmark and requests compensation for the non-hedgeable risk. We show that if the replication error is measured relative to the firm's value, the resulting formula is arbitrage free with mild restrictions on the parameters. [ABSTRACT FROM AUTHOR]

We study an enhanced hysteretic control system, with primary and secondary queues and random batch service. When the primary queue down-crosses r, the server operates on two parallel lines, servicing them asynchronously until the primary line of remaining units is processed or the number of serviced secondary units is at least S, whichever comes first. The server then waits until the primary queue length reaches N (if needed) before returning to primary service. The server capacity of primary units is limited by R with two options: r ≤ R ≤ N and R > N. Using fluctuation analysis we obtain closed-form distributions of available units during key periods of time and the steady state distribution of the primary queue. We illustrate analytical tractability by numerous analytical and computational examples. [ABSTRACT FROM AUTHOR]

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes. [ABSTRACT FROM AUTHOR]

We explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes. [ABSTRACT FROM AUTHOR]

In this paper we discuss the average of a Lévy process with a marginal negative-binomial distribution taken over a finite time interval, and simultaneously introduce a new class of absolutely continuous distribution based on Lerch's transcendent. Various distribution formulas are obtained in explicit form, including characteristic functions, distribution functions and moments. Some interesting asymptotics are also analyzed. As a consequence, we obtain rapidly converging series representations for the probability distribution of the average process. Numerical examples are provided in order to illustrate the proposed formulas. [ABSTRACT FROM AUTHOR]

In this paper we present a simple microscopic stochastic model describing short term plasticity within a large homogeneous network of interacting neurons. Each neuron is represented by its membrane potential and by the residual calcium concentration within the cell at a given time. Neurons spike at a rate depending on their membrane potential. When spiking, the residual calcium concentration of the spiking neuron increases by one unit. Moreover, an additional amount of potential is given to all other neurons in the system. This amount depends linearly on the current residual calcium concentration within the cell of the spiking neuron. In between successive spikes, the potentials and the residual calcium concentrations of each neuron decrease at a constant rate. We show that in this framework, short time memory can be described as the tendency of the system to keep track of an initial stimulus by staying within a certain region of the space of configurations during a short but macroscopic amount of time before finally being kicked out of this region and relaxing to equilibrium. The main technical tool is a rigorous justification of the passage to a large population limit system and a thorough study of the limit equation. [ABSTRACT FROM AUTHOR]