Limit Theorems For Additive Functionals Research Articles

Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Stable limit theorems for additive functionals of one-dimensional diffusion processes

Stable limit theorems for additive functionals of one-dimensional diffusion processes

Limit Theorems for Additive Functionals of Continuous-Time Lattice Random Walks in a Stationary Random Environment

For a continuous-time lattice random walk $X^\Lambda=\set{X^\Lambda_t,t\ge 0}$ in a random environment $\Lambda$, we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X^\Lambda_s)ds$, $t\ge 0$. We establish a limit theorem for it, which is similar to that in the non-lattice case, under less restrictive assumptions.

Limit theorems for additive functionals of the fractional Brownian motion

Limit theorems for additive functionals of the fractional Brownian motion

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.

Functional limit theorems for Volterra processes and applications to homogenization* *Johann Gehringer is supported by an EPSRC studentship, Xue-Mei Li by the EPRSC grants EP/V026100/1 and EP/S023925/1, and Julian Sieber by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process in the rough path topology. As an application, we establish weak convergence as ɛ → 0 of the solution of the random ordinary differential equation (ODE) and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type Itô SDE dx t = G(x t , dt), where G(x, t) is a semi-martingale with spatial parameters.

Central limit theorems for additive functionals and fringe trees in tries

We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in a random trie. Formulas for asymptotic mean and variance are given. In particular, the proportion of fringe trees of size k (defined as number of keys) is asymptotically, ignoring oscillations, c∕(k(k−1)) for k≥2, where c=1∕(1+H) with H the entropy of the letters. Another application gives asymptotic normality of the number of k-protected nodes in a random trie. For symmetric tries, it is shown that the asymptotic proportion of k-protected nodes (ignoring oscillations) decreases geometrically as k→∞.

Limit theorems for additive functionals of stochastic functional differential equations with infinite delay

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales.

Finite-memory elephant random walk and the central limit theorem for additive functionals

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well-known result with a long history. In this paper, we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from Gut and Stadtmüeller (2018). We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

On the local limit theorems for psi-mixing Markov chains

In this paper we investigate the local limit theorem for additive functionals of a nonstationary Markov chain with finite or infinite second moment. The moment conditions are imposed on the individual summands and the weak dependence structure is expressed in terms of some uniformly mixing coefficients.

Limit theorems for additive functionals of random walks in random scenery

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n1 4). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n1 8).

Limit theorems for additive functionals of continuous time random walks

AbstractFor a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$, t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

We construct P(ϕ)1-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for additive functionals of these processes. We discuss a number of examples by choosing specific functionals related to particle-field operators.

A new CLT for additive functionals of Markov chains

In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with var(Sn)∕n uniformly bounded, satisfies a n−central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stronger forms of ergodicity. In absence of ergodicity the convergence in distribution still holds, but the limiting distribution might not be normal.

Limit theorems for additive functionals of path-dependent SDEs

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

A Central Limit Theorem for Almost Local Additive Tree Functionals

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are "almost local" in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger.

Limit theorems related to the integral functionals of one dimensional fractional Brownian motion

In this paper, using the analog occupation time formulae associated with the integral functionals of one-dimensional fractional Brownian motion, we establish limit theorems for additive functionals for suitable functions f, where is a fractional Brownian motion.

A central limit theorem for additive functionals of increasing trees

AbstractA tree functional is called additive if it satisfies a recursion of the form$F(T) = \sum_{j=1}^k F(B_j) + f(T)$, whereB1, …,Bkare the branches of the treeTandf(T) is a toll function. We prove a general central limit theorem for additive functionals ofd-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group ofd-ary increasing trees, but other examples (old and new) are covered as well.

Potential kernel, hitting probabilities and distributional asymptotics

$\mathbb{Z}^{d}$-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

Functional Central Limit Theorem for Additive Functionals Associated to the Generalized Nelson Hamiltonian

Functional Central Limit Theorem for Additive Functionals Associated to the Generalized Nelson Hamiltonian