We characterize the equilibrium behavior of a class of stochastic particle systems, where particles (representing customers, jobs, animals, molecules, etc.) enter a space randomly through time, interact, and eventually leave. The results are useful for analyzing the dynamics of randomly evolving systems including spatial service systems, species populations, and chemical reactions. Such models with interactions arise in the study of species competitions and systems where customers compete for service (such as wireless networks). The models we develop are space-time measure-valued Markov processes. Specifically, particles enter a space according to a space-time Poisson process and are assigned independent and identically distributed attributes. The attributes may determine their movement in the space, and whenever a new particle arrives, it randomly deletes particles from the system according to their attributes. Our main result establishes that spatial Poisson processes are natural temporal limits for a large class of particle systems. Other results include the probability distributions of the sojourn times of particles in the systems, and probabilities of numbers of customers in spatial polling systems without Poisson limits.

A point process is a mechanism, which realizes randomly locally finite point measures. One of the main results of this thesis is an existence theorem for a new class of point processes with a so called signed Levy pseudo measure L, which is an extension of the class of infinitely divisible point processes. The construction approach is a combination of the classical point process theory, as developed by Kerstan, Matthes and Mecke, with the method of cluster expansions from statistical mechanics. Here the starting point is a family of signed Radon measures, which defines on the one hand the Levy pseudo measure L, and on the other hand locally the point process. The relation between L and the process is the following: this point process solves the integral cluster equation determined by L. We show that the results from the classical theory of infinitely divisible point processes carry over in a natural way to the larger class of point processes with a signed Levy pseudo measure. In this way we obtain e.g. a criterium for simplicity and a characterization through the cluster equation, interpreted as an integration by parts formula, for such point processes. Our main result in chapter 3 is a representation theorem for the factorial moment measures of the above point processes. With its help we will identify the permanental respective determinantal point processes, which belong to the classes of Boson respective Fermion processes. As a by-product we obtain a representation of the (reduced) Palm kernels of infinitely divisible point processes. In chapter 4 we see how the existence theorem enables us to construct (infinitely extended) Gibbs, quantum-Bose and polymer processes. The so called polymer processes seem to be constructed here for the first time. In the last part of this thesis we prove that the family of cluster equations has certain stability properties with respect to the transformation of its solutions. At first this will be used to show how large the class of solutions of such equations is, and secondly to establish the cluster theorem of Kerstan, Matthes and Mecke in our setting. With its help we are able to enlarge the class of Polya processes to the so called branching Polya processes. The last sections of this work are about thinning and splitting of point processes. One main result is that the classes of Boson and Fermion processes remain closed under thinning. We use the results on thinning to identify a subclass of point processes with a signed Levy pseudo measure as doubly stochastic Poisson processes. We also pose the following question: Assume you observe a realization of a thinned point process. What is the distribution of deleted points? Surprisingly, the Papangelou kernel of the thinning, besides a constant factor, is given by the intensity measure of this conditional probability, called splitting kernel. / Ein Punktprozess ist ein Mechanismus, der zufällig ein lokalendliches Punktmaß realisiert. Ein Hauptresultat dieser Arbeit ist ein Existenzsatz für eine sehr große Klasse von Punktprozessen mit einem signierten Levy Pseudomaß L. Diese Klasse ist eine Erweiterung der Klasse der unendlich teilbaren Punktprozesse. Die verwendete Methode der Konstruktion ist eine Verbindung der klassischen Punktprozesstheorie, wie sie von Kerstan, Matthes und Mecke ursprünglich entwickelt wurde, mit der sogenannten Methode der Cluster-Entwicklungen aus der statistischen Mechanik. Ausgangspunkt ist eine Familie von signierten Radonmaßen. Diese definiert einerseits das Levysche Pseudomaß L; andererseits wird mit deren Hilfe der Prozess lokal definiert. Der Zusammenhang zwischen L und dem Prozess ist so, dass der Prozess die durch L bestimmte Integralgleichung (genannt Clustergleichung) löst. Wir zeigen, dass sich die Resultate aus der klassischen Theorie der unendlich teilbaren Punktprozesse auf natürliche Weise auf die neue Klasse der Punktprozesse mit signiertem Levy Pseudomaß erweitern lassen. So erhalten wir z.B. ein Kriterium für die Einfachheit und eine Charackterisierung durch die Clustergleichung für jene Punktprozesse. Unser erstes Hauptresultat in Kapitel 3 zur Analyse der konstruierten Prozesse ist ein Darstellungssatz der faktoriellen Momentenmaße. Mit dessen Hilfe werden wir die permanentischen respektive determinantischen Punktprozesse, die in die Klasse der Bosonen respektive Fermionen Prozesse fallen, identifizieren. Als ein Nebenresultat erhalten wir eine Darstellung der (reduzierten) Palm Kerne von unendlich teilbaren Punktprozessen. Im Kapitel 4 konstruieren wir mit Hilfe unseres Existenzsatzes unendlich ausgedehnte Gibbsche Prozesse sowie Quanten-Bose und Polymer Prozesse. Unseres Wissens sind letztere bisher nicht konstruiert worden. Im letzten Teil der Arbeit zeigen wir, dass die Familie der Clustergleichungen gewisse Stabilitätseigenschaften gegenüber gewissen Transformationen ihrer Lösungen aufweist. Dies wird erstens verwendet, um zu verdeutlichen, wie groß die Klasse der Punktprozesslösungen einer solchen Gleichung ist. Zweitens wird damit der Ausschauerungssatz von Kerstan, Matthes und Mecke in unserer allgemeineren Situation gezeigt. Mit seiner Hilfe können wir die Klasse der Polyaschen Prozesse auf die der von uns genannten Polya Verzweigungsprozesse vergrößern. Der letzte Abschnitt der Arbeit beschäftigt sich mit dem Ausdünnen und dem Splitten von Punktprozessen. Wir beweisen, dass die Klassen der Bosonen und Fermionen Prozesse abgeschlossen unter Ausdünnung ist. Die Ergebnisse über das Ausdünnen verwenden wir, um eine Teilklasse der Punktprozesse mit signiertem Levy Pseudomaß als doppelt stochastische Poissonsche Prozesse zu identifizieren. Wir stellen uns auch die Frage: Angenommen wir beobachten eine Realisierung einer Ausdünnung eines Punktprozesses. Wie sieht die Verteilung der gelöschten Punktkonfiguration aus? Diese bedingte Verteilung nennen wir splitting Kern, und ein überraschendes Resultat ist, dass der Papangelou-Kern der Ausdünnung, abgesehen von einem konstanten Faktor, gegeben ist durch das Intensitätsmaß des splitting Kernes.

The modelling of biological communities is important to further the understanding of species coexistence and the mechanisms involved in maintaining biodiversity. This involves considering not only interactions between individual biological organisms, but also the incorporation of covariate information, if available, in the modelling process. This thesis explores the use of point processes to model interactions in bivariate point patterns within a Bayesian framework, and, where applicable, in conjunction with covariate data. Specifically, we distinguish between symmetric and asymmetric species interactions and model these using appropriate point processes. In this thesis we consider both pairwise and area interaction point processes to allow for inhibitory interactions and both inhibitory and attractive interactions. It is envisaged that the analyses and innovations presented in this thesis will contribute to the parsimonious modelling of biological communities.

A common interest of scientists in many fields is to understand the relationship between the dynamics of a physical system and the occurrences of discrete events within such physical system. Seismologists study the connection between mechanical vibrations of the Earth and the occurrences of earthquakes so that future earthquakes can be better predicted. Astrophysicists study the association between the oscillating energy of celestial regions and the emission of photons to learn the Universe's various objects and their interactions. Neuroscientists study the link between behavior and the millisecond-timescale spike patterns of neurons to understand higher brain functions. Such relationships can often be formulated within the framework of state-space models with point process observations. The basic idea is that the dynamics of the physical systems are driven by the dynamics of some stochastic state variables and the discrete events we observe in an interval are noisy observations with distributions determined by the state variables. This thesis proposes several new methodological developments that advance the framework of state-space models with point process observations at the intersection of statistics and neuroscience. In particular, we develop new methods 1) to characterize the rhythmic spiking activity using history-dependent structure, 2) to model population spike activity using marked point process models, 3) to allow for real-time decision making, and 4) to take into account the need for dimensionality reduction for high-dimensional state and observation processes. We applied these methods to a novel problem of tracking rhythmic dynamics in the spiking of neurons in the subthalamic nucleus of Parkinson's patients with the goal of optimizing placement of deep brain stimulation electrodes. We developed a decoding algorithm that can make decision in real-time (for example, to stimulate the neurons or not) based on various sources of information present in population spiking data. Lastly, we proposed a general three-step paradigm that allows us to relate behavioral outcomes of various tasks to simultaneously recorded neural activity across multiple brain areas, which is a step towards closed-loop therapies for psychological diseases using real-time neural stimulation. These methods are suitable for real-time implementation for content-based feedback experiments.

Recentemente, Guiol, Machado e Schinazi propuseram um modelo estocástico para a evolução de espécies. Nesse modelo, as intensidades de nascimentos de novas espécies e de ocorrências de extinções são invariantes ao longo do tempo. Ademais, no instante de nascimento de uma nova espécie, a mesma é rotulada com um número aleatório gerado de uma distribuição absolutamente contínua. Toda vez que ocorre uma extinção, apenas uma espécie morre - a com o menor número vinculado. Quando a intensidade com que surgem novas espécies é maior que a com que ocorrem extinções, existe um valor crítico f_c tal que todas as espécies rotuladas com números menores que f_c morrerão quase certamente depois de um tempo aleatório finito, e as rotuladas com números maiores que f_c terão probabilidades positivas de se tornarem perpétuas. No entanto, espécies menos aptas continuam a aparecer durante o processo evolutivo e não há a garantia do surgimento de uma espécie imortal. Consideramos um caso particular do modelo de Guiol, Machado e Schinazi e abordamos estes dois últimos pontos. Caracterizamos o processo pontual limite vinculado às espécies na fase subcrítica do modelo e discorremos sobre a existência de espécies imortais. / Recently, Guiol, Machado and Schinazi proposed a stochastic model for species evolution. In this model, births and deaths of species occur with intensities invariant over time. Moreover, at the time of birth of a new species, it is labeled with a random number sampled from an absolutely continuous distribution. Each time there is an extinction event, exactly one existing species disappears: that with the smallest number. When the birth rate is greater than the extinction rate, there is a critical value f_c such that all species that come with number less than f_c will almost certainly die after a finite random time, and those with numbers higher than f_c survive forever with positive probability. However, less suitable species continue to appear during the evolutionary process and there is no guarantee the emergence of an immortal species. We consider a particular case of Guiol, Machado and Schinazi model and approach these last two points. We characterize the limit point process linked to species in the subcritical phase of the model and discuss the existence of immortal species.

Recentemente, Guiol, Machado e Schinazi propuseram um modelo estocástico para a evolução de espécies. Nesse modelo, as intensidades de nascimentos de novas espécies e de ocorrências de extinções são invariantes ao longo do tempo. Ademais, no instante de nascimento de uma nova espécie, a mesma é rotulada com um número aleatório gerado de uma distribuição absolutamente contínua. Toda vez que ocorre uma extinção, apenas uma espécie morre - a com o menor número vinculado. Quando a intensidade com que surgem novas espécies é maior que a com que ocorrem extinções, existe um valor crítico f_c tal que todas as espécies rotuladas com números menores que f_c morrerão quase certamente depois de um tempo aleatório finito, e as rotuladas com números maiores que f_c terão probabilidades positivas de se tornarem perpétuas. No entanto, espécies menos aptas continuam a aparecer durante o processo evolutivo e não há a garantia do surgimento de uma espécie imortal. Consideramos um caso particular do modelo de Guiol, Machado e Schinazi e abordamos estes dois últimos pontos. Caracterizamos o processo pontual limite vinculado às espécies na fase subcrítica do modelo e discorremos sobre a existência de espécies imortais. / Recently, Guiol, Machado and Schinazi proposed a stochastic model for species evolution. In this model, births and deaths of species occur with intensities invariant over time. Moreover, at the time of birth of a new species, it is labeled with a random number sampled from an absolutely continuous distribution. Each time there is an extinction event, exactly one existing species disappears: that with the smallest number. When the birth rate is greater than the extinction rate, there is a critical value f_c such that all species that come with number less than f_c will almost certainly die after a finite random time, and those with numbers higher than f_c survive forever with positive probability. However, less suitable species continue to appear during the evolutionary process and there is no guarantee the emergence of an immortal species. We consider a particular case of Guiol, Machado and Schinazi model and approach these last two points. We characterize the limit point process linked to species in the subcritical phase of the model and discuss the existence of immortal species.

Simulation of rare events can be costly with respect to time and computational resources. For certain processes it may be more efficient to begin at the rare event and simulate a kind of reversal of the process. This approach is particularly well suited to reversible Markov processes, but holds much more generally. This more general result is formulated precisely in the language of stationary point processes, proven, and applied to some examples. An interesting question is whether this technique can be applied to Markov processes which are substochastic, i.e. processes which may die if a graveyard state is ever reached. First, some of the theory of substochastic processes is developed; in particular a slightly surprising result about the rate of convergence of the distribution pi(n) at time n of the process conditioned to stay alive to the quasi-stationary distribution, or Yaglom limit, is proved. This result is then verified with some illustrative examples. Next, it is demonstrated with an explicit example that on infinite state spaces the reversal approach to analyzing both the rate of convergence to the Yaglom limit and the likely path of rare events can fail due to transience.

No description available.

Using drones for cellular coverage enhancement is a recent technology that has shown a great potential in various practical scenarios. However, one of the main challenges that limits the performance of drone-enabled wireless networks is the limited flight time. In particular, due to the limited on-board battery size, the drone needs to frequently interrupt its operation and fly back to a charging station to recharge/replace its battery. In addition, the charging station might be responsible to recharge multiple drones. Given that the charging station has limited capacity, it can only serve a finite number of drones simultaneously. Hence, in order to accurately capture the influence of the battery limitation on the performance, it is required to analyze the dynamics of the time spent by the drones at the charging stations. In this thesis, we first use tools from queuing theory and stochastic geometry to study the influence of each of the charging stations limited capacity and spatial density on the performance of a drone-enabled wireless network. We then extend our work to rural areas where users are greatly impacted by low income, high cost of backhaul connectivity, and limited resources. Considering the limitation of the electricity supply scarcity in some rural regions, we investigate the possibility and performance enhancement of the deployment of renewable energy (RE) charging stations. We outline three practical scenarios, and use simulation results to demonstrate that RE charging stations can be a possible solution to address the limited on-board battery of UAVs in rural areas, specially when they can harvest and store enough energy.

This dissertation addresses a class of problems in point process theory called 'thinning'. By thinning we mean an operation whereby a point process is split into two point processes by some rule. We obtain the covariance structure between the thinned processes under various thinning rules. We first obtain this structure for independent Bernoulli thinning of an arbitrary point process. We show that if the point process is a renewal (stationary or ordinary) process, the thinned processes will be uncorrelated if and only if the renewal process is Poisson in which case the thinned processes are independent. Thus, we have a situation where zero correlation implies independence. We also show that while the intervals between events in the thinned processes may be uncorrelated, the counts need not be. Next, we obtain the covariance structure between the thinned processes resulting from a mark dependent thinning of a Markov renewal process with a Polish mark space. These results are used to study the overflow queue where we show that in equilibrium the input and overflow processes are uncorrelated as are the output and overflow processes. We thus provide an example where two uncorrelated but dependent renewal processes, neither of which is Poisson but which produce a Poisson process when superposed. Next, we study Bernoulli thinning of an alternating Markov process and show that the thinned process are uncorrelated if and only if the process is Poisson in which case the thinned processes are independent. Finally, we obtain the covariance structure obtained when a renewal process is thinned by an independent Markov chain. We show that if the renewal process is Poisson and the chain is stationary, the thinned processes will be uncorrelated if and only if the Markov chain is a Bernoulli process. We also show how to design the chain to obtain a pre-specified covariance function. We then close the dissertation by summarizing the work presented and indicating areas of further research including a short discussion of the use of Laplace functionals in the determination of joint distributions of thinned processes. / Ph. D.