Long strange segments,ruin probabilities and the effect of memory on moving average processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.     

White noise driven SPDEs with reflection: Existence,uniqueness and large deviation principles

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Diffusion processes on graphs: stochastic differential equations,large deviation principle

Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices. Received: 12 February 1997 / Revised version: 3 March 1999     

Poisson type approximations for the Markov binomial distribution

The Markov binomial distribution is approximated by the Poisson distribution with the same mean, by a translated Poisson distribution and by two-parametric Poisson type signed measures. Using an adaptation of Le Cam’s operator technique, estimates of accuracy are proved for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are obtained. For some auxiliary results, we used Stein’s method.     

Refinements of the Gibbs conditioning principle

Summary Refinements of Sanov's large deviations theorem lead via Csiszár's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence. Sharp bounds on the growth of the block length with the length of the conditioning sequence are derived.Partially supported by NSF DMS92-09712 grant and by a US-Israel BSF grantPartially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

A Central Limit Theorem for isotropic flows

We establish that the image of a measure, which satisfies a certain energy condition, moving under a standard isotropic Brownian flow will, when properly scaled, have an asymptotically normal distribution under almost every realization of the flow. We derive the same result for an initial point mass moved by an isotropic Kraichnan flow.     

Poisson approximation for the number of large digits of inhomogeneousf-expansions

We determine the exact rate of Poisson approximation and give a second-order Poisson-Charlier expansion for the number of excedances of a given levelL _n among the firstn digits of inhomogeneousf-expansions of real numbers in the unit interval. The application of this general result to homogeneousf-expansions and, in particular, to regular continued fraction expansions provides not only a generalization but also a strengthening of a classical Poisson limit theorem due to W. Doeblin.     

Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods. Received: 26 May 1998 / Revised version: 24 February 1999     

An asymptotic theory for sample covariances of Bernoulli shifts

Covariances play a fundamental role in the theory of stationary processes and they can naturally be estimated by sample covariances. There is a well-developed asymptotic theory for sample covariances of linear processes. For nonlinear processes, however, many important problems on their asymptotic behaviors are still unanswered. The paper presents a systematic asymptotic theory for sample covariances of nonlinear time series. Our results are applied to the test of correlations.     

Fluctuations of empirical means at low temperature for finite Markov chains with rare transitions in the general case

Summary. The integrated autocovariance and autocorrelation time are essential tools to understand the dynamical behavior of a Markov chain. We study here these two objects for Markov chains with rare transitions with no reversibility assumption. We give upper bounds for the autocovariance and the integrated autocorrelation time, as well as exponential equivalents at low temperature. We also link their slowest modes with the underline energy landscape under mild assumptions. Our proofs will be based on large deviation estimates coming from the theory of Wentzell and Freidlin and others [4, 3, 12], and on coupling arguments (see [6] for a review on the coupling method). Received 5 August 1996 / In revised form: 6 August 1997     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

A law of the iterated logarithm for heavy-tailed random vectors

For a random vector belonging to the (generalized) domain of operator semistable attraction of some nonnormal law we prove various variants of Chover's law of the iterated logarithm for the partial sum. Furthermore we also derive some large deviation results necessary for the proof of our main theorems. Received: 30 September 1998 / Revised version: 28 May 1999     

Large deviations for the radial processes of the Brownian motions on hyperbolic spaces

Using the heat kernel estimates by Davies (1989) and Anker et al. (1996), we show large deviations for the radial processes of the Brownian motions on hyperbolic spaces.     

A functional modulus of continuity for Brownian motion

In this paper, we prove a sharpening of large deviation for increments of Brownian motion in (p,r)-capacity and Hölder norm case. As an application, we obtain a functional modulus of continuity for (p,r)-capacity in the stronger topology.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.