A general non-convex large deviation result with applications to stochastic equations

We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of It? stochastic equations in R ^d driven by Brownian motion and a Poisson random measure. Received: 23 June 1999 / Revised version: 17 February 2000 / Published online: 22 November 2000     

Convex geometry of max-stable distributions

It is shown that max-stable random vectors in [0, ∞ ) ^d with unit Fréchet marginals are in one to one correspondence with convex sets K in [0, ∞ ) ^d called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function P ξ ≤ x of a max-stable random vector ξ with unit Fréchet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.      

Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Let X ₁ , X ₂ , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X _i ’s. In this paper, we study the asymptotics for the tail probability of the random sum S_N = ?_{k = 1}^N X_k {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X ₁  > x)), and the distribution function (d.f.) of X ₁ is dominatedly varying; (ii) P(X ₁  > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X ₁ and N are asymptotically comparable and dominatedly varying.     

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

On inhomogeneous diophantine approximation and Hausdorff dimension

Let Γ = Z A + Z ⁿ  ⊂ R ⁿ be a dense subgroup of rank n + 1 and let [^(w)] \hat{w} (A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥  [^(w)] \hat{w} (A), the Hausdorff dimension of the set B _v of points in R ⁿ that are v-approximable with respect to Γ is shown to be equal to 1/v.     

Probabilities of high excursions of Gaussian fields

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ(t) ≡ 0, D ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval T ì \mathbbR^m T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e^−Q  + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).     

On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ ^d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 2000     

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     

Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov

The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L ¹(ℝ/ℤ), then the averages

On the Hopfield model at the critical temperature

We study the Hopfield model at temperature 1, when thenumber M(N) of patterns grows a bit slower than N. We reach a goodunderstanding of the model whenever M(N)≤N/(log N)¹¹. For example, we show that if M(N)→∞, for two typical configurations σ ¹, σ ², (∑ _{i ≤ N} σ¹ _i σ² _i )² is close to NM(N). Received: 15 December 1999 / Revised version: 8 December 2000 / Published online: 23 August 2001     

The primes contain arbitrarily long polynomial progressions

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P ₁, …, P _k  ∈ Z[m] in one unknown m with P ₁(0) = … = P _k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with 1 \leqslant m \leqslant x^e1 \leqslant m \leqslant x^\varepsilon, such that x + P ₁(m), …, x + P _k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P _j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.     

Transversal fluctuations for increasing subsequences on the plane

Consider a realization of a Poisson process in ℝ² with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N ^χ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order N ^ξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models. Received: 16 April 1999 / Revised version: 5 July 1999 / Published online: 14 February 2000     

Time hierarchies for cryptographic function inversion with advice

We prove a time hierarchy theorem for inverting functions computable in a slightly nonuniform polynomial time. In particular, we prove that if there is a strongly one-way function, then for any k and for any polynomial p, there is a function f computable in linear time with one bit of advice such that there is a polynomial-time probabilistic adversary that inverts f with probability ≥ 1/p(n) on infinitely many lengths of input, while all probabilistic O(n ^k )-time adversaries with logarithmic advice invert f with probability less than 1/p(n) on almost all lengths of input. We also prove a similar theorem in the worst-case setting, i.e., if P ≠ NP, then for every l > k ≥ 1

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0.     

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝ ^d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form L_n[f](x): = ?_i=1ⁿ f_i(x)(f(y_i) + á?f(y_i), x-y_i?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y₁,..., y_n\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ ₁, ..., ϕ _n } is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L _n with a smallest constant in an L ^p error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator L _n for approximating f by L _n [f] with respect to the L ^p norm.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

Random Partial Orders Defined by Angular Domains

The d-dimensional random partial order is the intersection of d independently and uniformly chosen (with replacement) linear orders on the set [n] = {1, 2, . . . , n}. This is equivalent to picking n points uniformly at random in the d-dimensional unit cube Q_d=[0,1]^dQ_d=[0,1]^d with the coordinate-wise ordering. If d = 2, then this can be rephrased by declaring that for any pair P ₁, P ₂ ∈ Q ₂ we have P ₁ ≺ P ₂ if and only if P ₂ lies in the positive upper quadrant defined by the two axis-parallel lines crossing at P ₁. In this paper we study the random partial order with parameter α (0 ≤ α ≤ π) which is generated by picking n points uniformly at random from Q ₂ equipped with the same partial order as above but with the quadrant replaced by an angular domain of angle α.     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets

We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ ^ds/2 as λ→∞ where we can explicitly compute the spectral dimension d _s . Moreover the limit N(λ) λ ^−ds/2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal. Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000     

Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n ^−1/2 β _n log ^3/2 n in which β _n can be particularly taken as (log log n)^1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n ^−1/2(log log n)^1/2log ² n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n ^−1/2 β _n log ^1/2 n for the above β _n under the given covariance function, which improves the relevant one n ^−1/2(log log n)^1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.