Iterates of expanding maps

The iterates of expanding maps of the unit interval into itself have many of the properties of a more conventional stochastic process, when the expanding map satisfies some regularity conditions and when the starting point is suitably chosen at random. In this paper, we show that the sequence of iterates can be closely tied to an m-dependent process. This enables us to prove good bounds on the accuracy of Gaussian approximations. Our main tools are coupling and Stein's method. Received: 27 June 1997 / Revised version: 21 September 1998     

Some limit theorems for Hawkes processes and application to financial statistics

In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0,T]$[0,T]$ when T→∞$T\to \infty$. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh Δ$\Delta$ over [0,T]$[0,T]$ up to some further time shift τ$\tau$. The behaviour of this functional depends on the relative size of Δ$\Delta$ and τ$\tau$ with respect to T$T$ and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead–lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.     

Schilder theorem for the Brownian motion on the diffeomorphism group of the circle

We prove a large deviation principle for flows associated to stochastic differential equations with non-Lipschitz coefficients. As an application we establish a Schilder Theorem for the Brownian motion on the group of diffeomorphisms of the circle.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Rates of convergence in the central limit theorem for linear statistics of martingale differences

In this paper, we give rates of convergence for minimal distances between linear statistics of martingale differences and the limiting Gaussian distribution. In particular the results apply to the partial sums of (possibly long range dependent) linear processes, and to the least squares estimator in some parametric regression models.     

Limit theorems for hybridization reactions on oligonucleotide microarrays

We derive herein the limiting laws for certain stationary distributions of birth-and-death processes related to the classical model of chemical adsorption-desorption reactions due to Langmuir. The model has been recently considered in the context of a hybridization reaction on an oligonucleotide DNA-microarray. Our results imply that the truncated-gamma- and beta-type distributions can be used as approximations to the observed distributions of the fluorescence readings of the oligo-probes on a microarray. These findings might be useful in developing new model-based, probe-specific methods of extracting target concentrations from array fluorescence readings.     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

Asymptotic independence of maximum waiting times for increasing alphabet

For everyk≥1 consider the waiting time until each pattern of lengthk over a fixed alphabet of sizen appears at least once in an infinite sequence of independent, uniformly distributed random letters. Lettingn→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent. Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905.     

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x$x$ given by _μ1(x)→0${\mu }_1\left(x\right)\to 0$ as x→∞$x\to \infty$ is known as Lamperti’s problem  . We give sharp almost-sure bounds for processes of this type in the case where _μ1(x)${\mu }_1\left(x\right)$ is of order x^−β$x^{-\beta }$ for some β∈(0,1)$\beta \in (0,1)$. The bounds are of order t^1/(1+β)$t^{1/(1+\beta )}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)$(2+2\beta +\epsilon )$-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x^β_μ1(x)$x^{\beta }{\mu }_1\left(x\right)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0$\beta =0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.     

Asymptotic laws for regenerative compositions: gamma subordinators and the like

For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts K_n of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for K_n and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+. Research supported in part by N.S.F. Grant DMS-0071448     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.     

Generalized Hermite processes,discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0}$\{Z\left(t\right),t\ge 0\}$ called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1)$H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel g$g$, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g$g$ can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}$\left\{X\left(n\right)\right\}$. In addition, we consider a fractionally-filtered version Z^β(t)$Z^{\beta }\left(t\right)$ of Z(t)$Z\left(t\right)$, which allows H∈(0,1/2)$H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.     

Asymptotic normality of the sample mean and covariances of evanescent fields in noise

We consider the asymptotic properties of the sample mean and the sample covariance sequence of a field composed of the sum of a purely indeterministic and evanescent components. The asymptotic normality of the sample mean and sample covariances is established. A Bartlett-type formula for the asymptotic covariance matrix of the sample covariances of this field, is derived.     

Maximal words connected with unimodal maps

Orbits under a unimodal function are commonly characterized by letters of the alphabet (L, C, R). We present an algorithm for recognizing maximality of a word and study the necessary extension of the space of admissible words.