Functional Limit Theorem Research Articles

Functional Convergence of Berry’s Nodal Lengths: Approximate Tightness and Total Disorder

We consider Berry’s random planar wave model (J Phys A 10(12):2083–2092, 1977), and prove spatial functional limit theorems—in the high-energy limit—for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zitikis (Ann Inst Stat Math 60(2):345–365, 2008).

Limit theorems for additive functionals of the fractional Brownian motion

Limit theorems for additive functionals of the fractional Brownian motion

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Let ξ1, ξ2,… be i.i.d. random variables of zero mean and finite variance and η1, η2,… positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, α∈(0,1). The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump ξk occurs; if the present position of the Markov chain is nonpositive, then the jump ηk occurs. We prove functional limit theorems for this and two closely related Markov chains under Donsker’s scaling. The weak limit is a nonnegative process (X(t))t≥0 satisfying a stochastic equation dX(t)=dW(t)+dUα(LX(0)(t)), where W is a Brownian motion, Uα is an α-stable subordinator which is independent of W, and LX(0) is a local time of X at 0. Also, we explain that X is a Feller Brownian motion with a ‘jump-type’ exit from 0.

The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes.

A functional stable limit theorem for Gibbs–Markov maps

Pour une classe d’observables d-dimensionnelles localement (mais pas nécessairement uniformément) Lipschitz sur un système Gibbs–Markov, nous montrons que la convergence des sommes ergodiques (convenablement centrées et normalisées) vers un vecteur aléatoire de loi stable non gaussienne est équivalente au fait que la distribution appartient au domaine d’attraction classique. Dans ce cas nous montrons aussi un principe d’invariance faible dans la topologie forte de Skorohod J1 sur D([0,∞),Rd). L’argument utilise l’approche classique via les lois marginales de dimension finie et la tension de la suite dans le bon espace. Comme applications, nous démontrons une loi arcsinus à la Spitzer pour certaines Z-extensions des systèmes Gibbs–Markov, ainsi qu’une propriété d’indépendance asymptotique des processus d’excursion de certaines applications intermittentes de l’intervalle.

Historical Lattice Trees

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions d>8 converge to historical Brownian motion. This is a functional limit theorem for measure-valued processes that encodes the genealogical structure of the underlying random trees. Our results are applied elsewhere to prove that random walks on lattice trees, appropriately rescaled, converge to Brownian motion on super-Brownian motion.

Limit theorems for discounted convergent perpetuities II

Let (ξ1,η1), (ξ2,η2),… be independent identically distributed R2-valued random vectors. Assuming that ξ1 has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of η1 we prove three functional limit theorems for the logarithm of convergent discounted perpetuities ∑k≥0eξ1+…+ξk−akη k+1 as a→0+. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

A functional limit theorem for lattice oscillating random walks

A functional limit theorem for lattice oscillating random walks

Multi-Patch Epidemic Models with General Exposed and Infectious Periods

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The migration processes among the patches in either of the four states are assumed to be Markovian, and independent of the exposed and infectious periods. These periods have general distributions, and are not affected by the possible migrations of the individuals. The infection “from distance” aspect introduces a new formulation of the infection process, which, together with the migration processes, brings technical challenges in proving the functional limit theorems. Generalizing the methods in Pang and Pardoux [Ann. Appl. Probab. 32 (2022) 1615–1665], we establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

Asymptotics of a time bounded cylinder model

AbstractOne way to model telecommunication networks are static Boolean models. However, dynamics such as node mobility have a significant impact on the performance evaluation of such networks. Consider a Boolean model in $\mathbb {R}^d$ and a random direction movement scheme. Given a fixed time horizon $T>0$, we model these movements via cylinders in $\mathbb {R}^d \times [0,T]$. In this work, we derive central limit theorems for functionals of the union of these cylinders. The volume and the number of isolated cylinders and the Euler characteristic of the random set are considered and give an answer to the achievable throughput, the availability of nodes, and the topological structure of the network.

Tempered functional time series

We propose a broad class of models for time series of curves (functions) that can be used to quantify near long‐range dependence or near unit root behavior. We establish fundamental properties of these models and rates of consistency for the sample mean function and the sample covariance operator. The latter plays a role analogous to sample cross‐covariances for multivariate time series, but is far more important in the functional setting because its eigenfunctions are used in principal component analysis, which is a major tool in functional data analysis. It is used for dimension reduction of feature extraction. We also establish a central limit theorem for functions following our model. Both the consistency rates and the normalizations in the Central Limit Theorem (CLT) are nonstandard. They reflect the local unit root behavior and the long memory structure at moderate lags.

Local limit theorems for complex functions on [formula omitted

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the d-dimensional integer lattice, Zd. Under certain mild assumptions on the distribution, the theorem says that the convolution powers are well-approximated by a single scaled Gaussian density which we call an attractor. When such distributions are allowed to take on complex values, their convolution powers exhibit new and striking behaviors not seen in the probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville, P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a complete description of local limit theorems for the class of finitely supported complex-valued functions on Z. For convolution powers of complex-valued functions on Zd, much less is known. In a previous work by the author and L. Saloff-Coste, local limit theorems were established for complex-valued functions whose Fourier transform is maximized in absolute value at so-called points of positive homogeneous type and, in that case, the resultant attractors are generalized heat kernels corresponding to a class of higher order partial differential operators. By considering the possibility that the Fourier transform can be maximized in absolute value at points of imaginary homogeneous type, a notion motivated by V. Thomée, this article extends previous work of the author and L. Saloff-Coste to broaden the class of complex-valued functions for which it is possible to obtain local limit theorems. These local limit theorems contain attractors given by certain oscillatory integrals and their convergence is established using a generalized polar-coordinate integration formula, due to H. Bui and the author, and the Van der Corput lemma. The article also extends recent results on sup-norm type estimates of H. Bui and the author.

Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities

We consider a sequence of i.i.d. random variables, (ξ)=(ξi)i=0,1,2,⋯, Eξ0=0, Eξ02=1, and subordinate it by a doubly stochastic Poisson process Π(λt), where λ≥0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ(t)=ξΠ(λt) is known as the PSI-process. Elements of the triplet (Π,λ,(ξ)) are supposed to be independent. For sums of n, independent copies of such processes, normalized by n, we establish a functional limit theorem in the Skorokhod space D[0,T], for any T>0, under the assumption E|ξ0|2h<∞ for some h>1/γ2. Here, γ∈(0,1] reflects the tail behavior of the distribution of λ, in particular, γ≡1 when Eλ<∞. The limit process is a stationary Gaussian process with the covariance function Ee−λu, u≥0. As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales.

LOCAL INVARIANCE PRINCIPLE FOR A RANDOM WALK WITH ZERO DRIFT

We consider a random walk \(\left\{ S_{n},\text { }n\ge 0\right\}\) with zero drift and finite variance \(\sigma ^{2}\). Let T be the first hitting time of the semi-axis \(\left( -\infty ,0\right]\) by this random walk. For the random process \(\left\{ S_{\left\lfloor nt\right\rfloor }/\sigma \sqrt{n}{\small ,}\text { }t\in \left[ 0,1\right] \right\}\), considered under the condition that \(T=n\), a functional limit theorem on its convergence to the Brownian excursion, as \(n\rightarrow \infty\), is proved.

Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (Ann. Appl. Probab. 17 (2007) 1538–1569), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from Adamczak and Miłoś (Electron. J. Probab. 20 (2015) 42), where the evolution of the trait is given by an Ornstein–Uhlenbeck process.

Joint Invariance Principles for Random Walks with Positively and Negatively Reinforced Steps

Given a random walk (S_n) with typical step distributed according to some fixed law and a fixed parameter p in (0,1), the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability 1-p, the same step as (S_n) while with probability p, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when p < 1/2 and when the typical step is centred. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, B(t),tp∫0ts-pdBr(s),t-p∫0tspdBc(s)t∈R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( B(t) , \\, t^p \\int _0^t s^{-p} \\mathrm {d}B^r(s) , \\, t^{-p} \\int _0^t s^{p} \\mathrm {d}B^c(s) \\right) _{t \\in \\mathbb {R}^+} \\end{aligned}$$\\end{document}for properly correlated Brownian motions B, B^r, B^c. The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes p in (0,1) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.

Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations

AbstractLet $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions

A nested Karlin’s occupancy scheme is a symbiosis of classical Karlin’s balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer j, identify the jth generation individuals with the jth generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights.Assume that there are n balls and that the discrete probability distribution (pk)k∈N responsible for the first generation is Weibull-like. This means that the counting function of 1/p1, 1/p2,… belongs to de Haan’s class Π, a subclass of the class of slowly varying functions. Denote by Kn(j)(l) and Kn∗(j)(l) the number of the jth generation boxes which contain at least l balls and exactly l balls, respectively. We prove functional limit theorems (FLTs) for the matrix-valued processes (K[eT+⋅](j)(l))j,l∈N and (K[eT+⋅]∗(j)(l))j,l∈N, properly normalized and centered, as T→∞. The present FLTs are an extension of a FLT proved by Iksanov et al. (2022) for the vector-valued process (K[eT+⋅](j)(1))j∈N. While the rows of each of the limit matrix-valued processes are independent and identically distributed, the entries within each row are stationary Gaussian processes with explicitly given covariances and cross-covariances. We provide an integral representation for each row. The results obtained are new even for Karlin’s occupancy scheme.

Spatial integral of the solution to hyperbolic Anderson model with time-independent noise

In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension d≤2, as the domain of the integral gets large (for fixed time t). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô’s martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein’s method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.

Central limit theorem for bifurcating markov chains under L2-ergodic conditions

AbstractBifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under $L^2$ -ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.