Precise large deviation estimates for a one-dimensional random walk in a random environment

ω_x } (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when at x, moves one step to the right with probability ω _x , and one step to the left with probability 1 −ωx. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a “coarse graining scheme” together with comparison techniques. Received: 22 July 1997/Revised version: 15 June 1998     

Large deviation principles for Euclidean functionals and other nearly additive processes

We prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or Euclidean space, under approximate additivity and regularity hypotheses. The rate function is the convex dual of the limiting logarithmic moment generating function. In some applications the rate function can be expressed in terms of relative entropy. The general result applies to processes in Euclidean combinatorial optimization, statistical mechanics, and computational geometry. Examples include the length of the minimal tour (the traveling salesman problem), the length of the minimal matching graph, the length of the minimal spanning tree, the length of the k-nearest neighbors graph, and the free energy of a short-range spin glass model. Received: 3 April 1999 / Revised version: 23 June 1999 / Published online: 8 May 2001     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997     

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     

One-sided local large deviation and renewal theorems in the case of infinite mean

Summary. If {S _n ,n≧0} is an integer-valued random walk such that S _n /a _n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S _n =r} for values of r such that r/a _n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a _n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S _n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems. Received: 9 August 1995 / In revised form: 29 September 1996     

A polynomial of graphs on surfaces

ribbon graphs , i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a four-variable polynomial invariant of these objects, the ribbon graph polynomial, which has all the main properties of the Tutte polynomial. Although the ribbon graph polynomial extends the Tutte polynomial, its definition is very different, and it depends on the topological structure in an essential way. Received: 14 September 2000 / Published online: 18 January 2002     

Laplace approximations for sums of independent random vectors

Let X _i , i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping B→R. Under a central limit theorem assumption, an asymptotic evaluation of Z _n = E (exp (n Φ (∑ _{i =1} ⁿ X _i /n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without the central limit theorem assumption. Received: 19 September 1997 / Revised version:22 April 1999     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

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     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems

Summary. Let (X _t ,t∈Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study quadratic forms of bivariate Appell polynomials of the sequence (X _t ) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution. We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple Wiener-It? integrals involving correlated Gaussian measures. Received: 22 August 1996 / In revised form: 30 August 1997     

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

On signed normal-Poisson approximations

For lattice distributions a convolution of two signed Poisson measures proves to be an approximation comparable with the normal law. It enables to get rid of cumbersome summands in asymptotic expansions and to obtain estimates for all Borel sets. Asymptotics can be constructed two-ways: by adding summands to the leading term or by adding summands in its exponent. The choice of approximations is confirmed by the Ibragimov-type necessary and sufficient conditions. Received: 20 November 1996 / Revised version: 5 December 1997     

Rates of convergence of random walk on distance regular graphs

When run on any non-bipartite q-distance regular graph from a family containing graphs of arbitrarily large diameter d, we show that d steps are necessary and sufficient to drive simple random walk to the uniform distribution in total variation distance, and that a sharp cutoff phenomenon occurs. For most examples, we determine the set on which the variation distance is achieved, and the precise rate at which it decays. The upper bound argument uses spectral methods – combining the usual Cauchy-Schwarz bound on variation distance with a bound on the tail probability of a first-hitting time, derived from its generating function. Received: 2 April 1997 / Revised version: 10 May 1998     

Normal limit theorems for symmetric random matrices

Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O _n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O ^k _n tends to a Brownian motion as n→∞. Received: 3 February 1998 / Revised version: 11 June 1998     

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     

Computing the generalized singular values/vectors of large sparse or structured matrix pairs

Summary. We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair . The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair . Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm. Received April 1, 1994 / Revised version received December 15, 1994     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Large deviations for increasing sequences on the plane

We prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among Poisson points on the plane. The rate function for lower tail deviations is derived from a 1977 result of Logan and Shepp about Young diagrams of random permutations. For the upper tail we use a coupling with Hammersley's particle process and convex-analytic techniques. Along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric Hammersley's process. Received: 22 July 1997 / Revised version: 23 March 1998