Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

A Necessary and Sufficient Condition for the Existence of Some Ternary [n, k, d] Codes Meeting the Greismer Bound

Let k and d be any integers such that k 4 and . Then there exist two integers and in {0,1,2} such that . The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary code meeting the Griesmer bound if and only if and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and , there is no projective ternary code for any integers k and such that 1k-3, where and for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results.     

A Characterization of Some Minihypers and its Application to Linear Codes

Any {f,r- 2+s; r,q}-minihyper includes a hyperplane in PG(r, q) if fr-1 + s 1 + q – 1 for 1 s q – 1, q 3, r 4, where i = (qi + 1 – 1)/ (q – 1 ). A lower bound on f for which an {f, r – 2 + 1; r, q}-minihyper with q 3, r 4 exists is also given. As an application to coding theory, we show the nonexistence of [ n, k, n + 1 – qk – 2 ]q codes for k 5, q 3 for qk – 1 – 2q – 1 < n qk – 1 – q – 1 when k > q – q - \sqrt q + 2$$ " align="middle" border="0"> and for when , which is a generalization of [18, Them. 2.4].     

Recurrence-Transience for Self-similar Additive Processes Associated with Stable Distributions

For any stable distribution on the line, recurrence-transience of the selfsimilar additive process {X _t ,t0} with (X ₁)= is determined. Comparison with the stable Lévy process {Y _t ,t0} with (Y ₁)= is made: if is not strictly stable, then {Y _t } is transient but {X _t } is recurrent except the obviously transient case of monotone sample functions.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {a _n,n1}, an array of rowwise independent and stochastically dominated random elements { V _nj, j1, n1} in a real separable Rademacher type p (1p2) Banach space, and a sequence of positive integer-valued random variables {T _n, n1}, a general weak law of large numbers of the form is established where {c _nj, j1, n1}, _n , b _n are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V _nj, j1, n1}. Illustrative examples include one wherein the strong law of large numbers fails.     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Differentiability of Inverse Operators and Limit Theorems for Inverse Functions

Let H be a map from a set SR ^d to R ^d . For tR ^d let _H (t) denote the distance from t to the set H(S). Consider sequences {s _n} _n1 in S such that . Any limit point of any such sequence (finite or infinite) is considered as a possible value of the inverse H ^–1(t). Any map defined in such a way will be called an SC-inverse (a selected closest inverse) to H. In the paper we study differentiability of the nonlinear operator at H=G, where G is a one-to-one map from S onto a set TR ^d with good analytic properties (specifically, a diffeomorphism). We establish compact differentiability of this operator tangentially to continuous functions and introduce a family of norms such that it is Fréchet differentiable with respect to them. We also obtain optimal bounds for the remainder of the differentiation, extending to the multivariate case recent results of Dudley. These differentiability results are applied to random maps , which could be statistical estimators of an unknown map G. For a function J on R ^d , let (J) _T be its restriction to T. It is shown that for a diffeomorphism G and for an increasing sequence of positive numbers {a _n } _n1 weak convergence of the sequence {a _n (G _n – G)} _n1 (locally in S) is equivalent to weak convergence of the sequence (locally in T) along with the convergence of the sequence to 0 in probability (locally uniformly in S). The equivalence holds for all SC-inverses and all double SC-inverses and it extends to the multivariate case a theorem of Vervaat. Moreover, each of these equivalent statements implies a kind of Taylor expansion of the SC-inverse at G (locally uniformly in T) where inv(A) denotes the inverse of a nonsingular linear transformation A in R ^d . Such limit theorems for functional inverses can be used to study asymptotic behavior of statistical estimators defined implicitly (as solutions of equations involving the empirical distribution P _n ). We show how to apply this approach to get asymptotic normality of M-estimators in the multivariate case under minimal assumptions. We consider an extension of the quantile function to the multivariate case related to M-parameters of a distribution P in R ^d (an M-quantile function)and use limit theorems for functional inverses to study limit behavior of the empirical M-quantile process. We also show how to use these theorems to study asymptotics of regression quantiles.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set (S) covers all G for eachs ubset SG\{0} of cardinality |S|s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p–2, thus establishing a conjecture of Diderrich.We characterize the critical sets with |S|=|G|/p+p–3 and (S)=G, where p3 is the smallest prime dividing n, n/p is composite and n7p²+3p.We also extend a result of Diderrichan d Mann by proving that, for n67, |S|n/3+2 and S=G imply (S)=G. Sets of cardinality for which (S) =G are also characterized when n183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality (G)=G to hold when |S|n/(p+2)+p, where p5, n/p is composite and n15p².* Work partially supported by the Spanish Research Council under grant TIC2000-1017 Work partially supported by the Catalan Research Council under grant 2000SGR00079     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Invariant subsets of strongly continuous semigroups

Let {P(t): t0} be a strongly continuous semigroup on a Banach space X and let |\| be a continuous norm on X such that |P(t)x|exp(t)|x|, XX, t0. Let C be a |\|-closed convex subset of X and suppose that for every x in D(A) there exists a sequence (x_n : n ) in D(A) with the following properties: lim|x–x_n|=0, lim|Ax–Ax_n|=0 and every x_n has a best approximation in C (with respect to |\|) which belongs to D(A). Then P(t)CC for all t0 if and only if, for every v in CD(A), the vector Av belongs to the |\|-closure of [0, ) (C-V).     

Presentations for Certain Chevalley Groups

Let G = P SL_n(K), n 3, K a division ring or D_n(K), n 4 or E_n(K), 6 n 8, K a field. Then two types of presentations for G are given. In the first, G is generated by SL₂(K)'s which satisfies relations according to the Dynkin diagram of G. In the second, G is generated by a set {A_i | i I } of Abelian groups, which satisfy relations similar to the root subgroups of G.     

For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?

Denote by the class of all triangle-free graphs on n vertices and m edges. Our main result is the following sharp threshold, which answers the question for which densities a typical triangle-free graph is bipartite. Fix > 0 and let . If n/2 m (1 – ) t ₃, then almost all graphs in are not bipartite, whereas if m (1 + )t ₃, then almost all of them are bipartite. For m (1 + )t ₃, this allows us to determine asymptotically the number of graphs in . We also obtain corresponding results for C -free graphs, for any cycle C of fixed odd length. Forschergruppe Algorithmen, Struktur, Zufall supported by Deutsche Forschungsgemeinschaft grant FOR 413/1-1     

Asymptotics and Bounds for Multivariate Gaussian Tails

Let {X_n, n 1} be a sequence of centered Gaussian random vectors in , d 2. In this paper we obtain asymptotic expansions (n ) of the tail probability P{X_n >t_n} with t_n a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.     

On the Equation xp + y q = z r : A Survey

Let p, q and r be three integers 2. During the last ten years many new ideas have emerged for the study of the Diophantine equation x + y = z. This study is divided into three parts according whether (p, q, r) = is negative, zero or positive. For instance, if we have (p, q, r) < 0, this study is closely connected to the theory of modular representations of dimension 2 in finite characteristic. The approach of associating to this equation Galois representations given by the division points of elliptic curves is very efficient. The aim of this paper is to present this circle of ideas and the known results concerning this equation.     

Continuous convolution semigroups integrating a submultiplicative function

Let be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure . It is shown that the measures of S integrate if and only if integrates outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate if and only if the random variable sup { (X(t)):0t1} is integrable (Theorem 2).     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.