Functional Law Of Large Numbers Research Articles

Superinfection and the hypnozoite reservoir for Plasmodium vivax: a general framework

A characteristic of malaria in all its forms is the potential for superinfection (that is, multiple concurrent blood-stage infections). An additional characteristic of Plasmodium vivax malaria is a reservoir of latent parasites (hypnozoites) within the host liver, which activate to cause (blood-stage) relapses. Here, we present a model of hypnozoite accrual and superinfection for P. vivax. To couple host and vector dynamics for a homogeneously-mixing population, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered by coupling expected host and vector dynamics (i.e. a hybrid approximation) or through a standard compartment modelling approach. Recognising that the subset of these equations that model the infection status of the human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter process to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a parameter R_0 expressible in terms of the primitives of the model, with the disease-free equilibrium shown to be uniformly asymptotically stable if R_0<1 and an endemic equilibrium solution emerging if R_0>1. Our work provides a theoretical basis to explore the epidemiology of P. vivax, and introduces a strategy for constructing tractable population-level models of malarial superinfection that can be generalised to allow for greater biological realism in a number of directions.

Asymptotic behavior of multiscale stochastic partial differential equations with Hölder coefficients

In this paper, we establish a quantified asymptotic analysis for a semi-linear slow-fast stochastic partial differential equation with Hölder coefficients. By studying the Poisson equation in Hilbert space, we first prove the strong convergence in the averaging principle, which is viewed as a functional law of large numbers. Then we study the stochastic fluctuations between the original system and its averaged equation. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type process, which is viewed as a functional central limit theorem. Rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergences are shown not to depend on the regularity of the coefficients in the equation for the fast variable, which coincides with the intuition, since in the limit systems the fast component has been totally averaged or homogenized out.

Small noise asymptotics of multi-scale McKean-Vlasov stochastic dynamical systems

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations.

Functional Law of Large Numbers and PDEs for Epidemic Models with Infection-Age Dependent Infectivity.

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time since infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic Volterra-type integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.

Functional law of large numbers and central limit theorem for slow-fast McKean-Vlasov equations

In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle of the functional law of large numbers type. In particular, the diffusion coefficient of the slow process can depend on the distribution of the fast motion. Then we consider the stochastic fluctuations of the original system around its average, and prove that the normalized difference will converge weakly to a linear McKean-Vlasov Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Extra drift and diffusion coefficients involving the expectation are characterized explicitly. Furthermore, the optimal rates of the convergence are also obtained.

Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime

We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, a general self-exciting function of event “ages” at each time and marks. The marks are assumed to be conditionally independent given the event times, while the distribution of each mark depends on the event time, that is, time-varying. We first observe an immigration–birth (branching) representation of such a non-stationary marked Hawkes process, and then derive an equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity gets large, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLTs) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLTs are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation. We also consider the special cases of multiplicative self-exciting functions, and an indicator type of non-decomposable self-exciting functions (including the cases of “ceasing” and “delayed” reproductions as well as their extensions with varying reproduction rates), and study the properties of the limiting Gaussian processes in these special cases.

Zeros of smooth stationary Gaussian processes

Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $\nu_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of $\nu_R$. Under suitable assumptions on the regularity of $f$ and the decay of its correlation function at infinity, we derive the asymptotics as $R \to +\infty$ of the central moments of the linear statistics of $\nu_R$. In particular, we derive an asymptotics of order $R^\frac{p}{2}$ for the $p$-th central moment of the number of zeros of $f$ in $[0,R]$. As an application, we derive a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures~$\nu_R$. More precisely, after a proper rescaling, $\nu_R$ converges almost surely towards the Lebesgue measure in weak-$*$ sense. Moreover, the fluctuation of $\nu_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the $k$-point function of the zero point process of~$f$, for any $k \geq 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this $k$-point function near any point of the large diagonal in~$\mathbb{R}^k$, thus quantifying the short-range repulsion between zeros of $f$. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of $f$.

Functional limit theorems for marked Hawkes point measures

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian motion results from the self-exciting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host as well as their interaction with that host.

Asymptotic analysis for affine point processes with large initial intensity

In this paper, we consider a generalized stochastic model associated with affine point processes based on several classical models. In particular, we study the asymptotic behavior of the process when the initial intensity is large, i.e. the intensity of arriving events observed initially is considerably larger, which appears in many real applications. For our generalized model, we establish (i) the large deviation principle; (ii) the corresponding functional law of large numbers; (iii) the corresponding central limit theorem, that reflect the fundamentals of the process asymptotic behavior. Our obtained results include existing results as special cases with a more general structure.

Scaling limits for sub-ballistic biased random walks in random conductances

We consider biased random walks in positive random conductances on the $d$-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional law of large numbers for the position of the walker, properly rescaled. Moreover, we state a functional central limit theorem where an atypical process, related to the fractional kinetics, appears in the limit.

Systems of interacting diffusions with partial annihilation through membranes

We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface $I$. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.

Randomly perturbed switching dynamics of a dc/dc converter

In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order DC/DC buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the ON/OFF configuration of the circuit, between two ordinary differential equations (ODE), and a discrete component which keeps track of the ON/OFF configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input in the ON state is subject to small white noise fluctuations of size \begin{document}$\varepsilon $\end{document} , \begin{document}$0 . For the ensuing stochastic system whose dynamics switch at random times between a small noise stochastic differential equation (SDE) and an ODE, we prove a functional law of large numbers which states that in the limit of vanishing noise, the stochastic system converges to the underlying deterministic one on time horizons of order \begin{document}$\mathscr{O}(1/\varepsilon ^ν)$\end{document} , \begin{document}$0 ≤ ν .

The jamming constant of uniform random graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a “parking process” on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given–possibly unbounded–degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.

G/G/∞ queues with renewal alternating interruptions

AbstractWe study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics:Xr(t,y) tracking the number of customers in the system at timetthat have residual service times strictly greater thany. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the processXr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻,L1)) endowed with the SkorokhodM1topology. The limit processes in the FCLT possess a stochastic decomposition property.

A scaling limit for the degree distribution in sublinear preferential attachment schemes

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non-tree’ evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Limits for density dependent time inhomogeneous Markov processes

A new functional law of large numbers to approximate a time inhomogeneous Markov process that is only density dependent in the limit as an index parameter goes to infinity is developed. This extends previous results by other authors to a broader class of Markov processes while relaxing some of the conditions required for those results to hold. This result is applied to a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics with the novel addition of time dependent dynamics. The resulting nonautonomous differential equation is analysed to provide conditions for extinction and persistence for a number of examples. This condition shows that the migration of a species will positively impact the reproduction in less populated areas while negatively impacting densely populated areas.

A modeling approach of the chemostat

Population dynamics and in particular microbial population dynamics, though intrinsically discrete and random, are conventionally represented as deterministic differential equations systems. In these type of models, populations are represented by continuous population sizes or densities usually with deterministic dynamics.Over the last decades, alternate individual-based models have been proposed where population is explicitly represented as a set of individuals. These may include stochastic dynamics or stochastic rules. With reference to the last class of models we can also associate pure jump processes where the population is described as a discrete population size with stochastic discrete event evolutions.In the first class of models the population dynamics and its representation may be viewed respectively as deterministic and continuous, in the second class they may be viewed respectively as stochastic and discrete.In this present work, we present a modeling approach that bridges the two representations. This link can be mathematically described as a functional law of large numbers in high population size asymptotics. These results suggest new strategies of modeling and simulation. We illustrate this approach on the modeling of the chemostat.

Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the population dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.

A model for a spatially structured metapopulation accounting for within patch dynamics

We develop a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics. Using a deterministic approximation derived from a functional law of large numbers, we develop conditions for extinction and persistence of the metapopulation in terms of the birth, death and migration parameters. Interestingly, we observe the Allee effect in a metapopulation comprising two patches of greatly different sizes, despite there being decreasing patch specific per-capita birth rates. We show that the Allee effect is due to the way the migration rates depend on the population density of the patches.

Fluid Limits for Overloaded Multiclass FIFO Single-Server Queues with General Abandonment

We consider an overloaded multiclass nonidling first-in-first-out single-server queue with abandonment. The interarrival times, service times, and deadline times are sequences of independent and identically, but generally distributed random variables. In prior work, Jennings and Reed studied the workload process associated with this queue. Under mild conditions, they establish both a functional law of large numbers and a functional central limit theorem for this process. We build on that work here. For this, we consider a more detailed description of the system state given by K finite, nonnegative Borel measures on the nonnegative quadrant, one for each job class. For each time and job class, the associated measure has a unit atom associated with each job of that class in the system at the coordinates determined by what are referred to as the residual virtual sojourn time and residual patience time of that job. Under mild conditions, we prove a functional law of large numbers for this measure-valued state descriptor. This yields approximations for related processes such as the queue lengths and abandoning queue lengths. An interesting characteristic of these approximations is that they depend on the deadline distributions in their entirety.