A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.     

Operator-semistable,operator semi-selfdecomposable probability measures and related nested classes on <Emphasis Type="Italic">p</Emphasis>-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body. (Received 25 January 2000)     

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 Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

We calculate E[V ₄(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find

The Typical Growth of the kth Excess in a Random Integer Partition

 For a partition , of a positive integer n chosen uniformly at random from the set of all such partitions, the kth excess is defined by if . We prove a bivariate local limit theorem for as . The whole range of possible values of k is studied. It turns out that ρ and η _k are asymptotically independent and both follow the doubly exponential (extreme value) probability law in a suitable neighbourhood of . Received February 6, 2001; in revised form February 25, 2002 Published online August 5, 2002     

On the smallest maximal increment of partial sums of i.i.d. random variables

Summary. We study the almost sure limiting behavior of the smallest maximal increment of partial sums of independent identically distributed random variables for a variety of increment sizes , where is a sequence of integers satisfying , and going to infinity at various rates. Our aim is to obtain universal results on such behavior under little or no assumptions on the underlying distribution function. Received: 30 August 1995 / In revised form: 27 September 1996     

A Normal Law for Matchings

Dedicated to the memory of Paul Erdős, both for his pioneering discovery of normality in unexpected places, and for his questions, some of which led (eventually) to the present work.   For a simple graph G, let be the size of a matching drawn uniformly at random from the set of all matchings of G. Motivated by work of C. Godsil [11], we give, for a sequence and , several necessary and sufficient conditions for asymptotic normality of the distribution of , for instance

On Sums of Units

It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in particular to rings of integers in finite extensions of the rationals. On the other hand there are many infinite algebraic extensions of the rationals in which every integer is a sum of two units.     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

The Penrose Polynomial of Binary Matroids

 A polynomial, called the Penrose polynomial, is studied for binary matroids, generalizing previous work on plane graphs. In particular, several formulae of Penrose are extended via the theory of isotropic systems. (Received 9 February, 2000; in revised form 26 June 2000)     

On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C ^2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C ^k . The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity. Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000     

On Patterns in Sequences of Random Events

 This paper considers patterns of particular events in sequences of trials, some independent and others Markovian. Matrix recursions are found for the number of sequences of length n avoiding a specific pattern, and the associated probability of this event is evaluated. A Markov chain method for the study of such problems is outlined, and is illustrated in various cases. Finally, configurations of length 3 in Bernoulli trials are examined as an example. Received 12 January 1998 in revised form 26 June 1998     

Fourier-Based Distances and Berry-Esseen Like Inequalities for Smooth Densities

 This paper is devoted to the rate of convergence problem in the central limit theorem for sums of independent identically distributed random variables with regular probability density function. The method we use depends strictly on Fourier based metrics, and yields Berry-Esseen like bounds for the convergence towards both a normal and a stable law in various Sobolev norms. Received 31 May 2001; in revised form 13 November 2001     

Inégalités de concentration pour les processus empiriques de classes de parties

We propose new concentration inequalities for maxima of set-indexed empirical processes. Our approach is based either on entropy inequalities or on martingale methods. The improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases. Received: 11 January 2000 / Revised version: 12 May 2000 / Published online: 14 December 2000     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Quasi-Diagonal Behaviour and Smooth Weyl Sums

 Estimates are provided for small moments of exponential sums over smooth numbers substantially sharper than available hitherto. These bounds arise from the author’s recent breaking of “classical convexity” in Waring’s problem. The methods underlying these new estimates provide guidance on good choices of parameters in the new iterative methods for smaller exponents. (Received 13 September 1999)     

On the Constants in the Meyer Inequality

 We study the growth of the constants in the Meyer inequality as p → 1 and p → ∞. Both constants grow, within constant factors, like (p − 1)⁻¹ and like p respectively. Received August 10, 2001; in revised form February 5, 2002     

Real Principal Value Integrals

 We study principal value integrals of multi-valued differential forms on compact spaces, as introduced by Langlands. Using resolutions of singularities we extend Langlands definition to the case in which the differential form may have no normal crossings. We show by an example that for non-compact spaces the principal value integral associated to a compactification may depend on the compactification. Principal value integrals appear as residues of poles of distributions and as coefficients of asymptotic expansions of oscillating integrals and fibre integrals. (Received 11 November 1999)