On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

On the L 1 convergence of kernel estimators of regression functions with applications in discrimination

Summary An estimate m _n of a regression function m(x)=E{Y|X=x} is weakly (strongly) consistent in L ₁ if ¦m _n (x)-m(x)¦(dx) converges to 0 in probability (w.p. 1) as the sample size grows large ( is the probability measure of X).We show that the well-known kernel estimate (Nadaraya, Watson) and several recursive modifications of it are weakly (strongly) consistent in L ₁ under no conditions on (X, Y) other than the boundedness of Y and the absolute continuity of . No continuity restrictions are put on the density corresponding to . We further notice that several kernel-type discrimination rules are weakly (strongly) Bayes risk consistent whenever X has a density.Research of both authors was sponsored by AFOSR Grant 77-3385     

A characterization of geometric ergodicity

Summary A homogeneous Markov chain on a countable state space can be classified as ergodic, geometrically ergodic, or strongly ergodic. Ergodicity and strong ergodicity have been characterized using the -coefficient. In this paper the -coefficient is used to characterize geometric ergodicity.     

On weakly mixing Markov operators and non-singular transformations

Summary Let P be a Markov operator on L (X, , m). Theorem 1: (i) P is weakly mixing (ii) For every fL there is a sequence {n_t} of density 1 such that all w ^*-cluster points of are constants (iii) For every fL there is a {k_j} with w ^*-convergent to a constant. Theorem 2: If P is induced by a non-singular transformation , P is weakly mixing For every A, { ^–n(A)} has a remotely trivial subsequence. The existence of a finite invariant measure is not required in these results.     

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

Minimal displacement of branching random walk

Summary Let denote a branching random walk in with mean particle productionm, m>1, and with incremental spatial distributionG, withG({0}) =p andG({1})=1–p. Ifmp=1, then the minimal displacement of behaves asymptotically like log logn/log 2. If the conditionG({1})=1–p is replaced byG((0, ))=1–p, we obtain a similar result.Research was partially supported by the National Science Foundation under grant MCS-7607039     

Evolution systems and perturbations of generators of strongly continuous groups

Suppose A generates a strongly continuous linear group on a Banach space X and B is a linear operator on X. It is shown that an extension of generates a strongly continuous semigroup if and only if the family of operators has an appropriate evolution system. This produces simple sufficient conditions for an extension of to generate a strongly continuous semigroup, including

Regular (2, 2)-systems

Let (E,I) be an independence system over the finite setE = {e ₁, ,e _n }, whose elements are orderede ₁ e _n . (E,I) is called regular, if the independence of {e _l , ,e _{l k} },l ₁ < <l _k , implies that of {e _{m l} , ,e _{m k} }, wherem _l < ··· <m _k andl ₁ m ₁, ,l _k m _k . (E,I) is called a 2-system, if for anyI I,e E I the setI {e } contains at most 2 distinct circuitsC, C I and the number 2 is minimal with respect to this property. If, in addition, for any two independent setsI andJ the family (C J, C C (J, I)), whereC(J, I) denotes {C C:e J I C {e}}, can be partitioned into 2 subfamilies each of which possesses a transversal, then (E,I) is called a (2, 2)-system. In this paper we characterize regular 2-systems and we show that the classes of regular 2-systems resp. regular (2, 2)-systems are identical.     

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     

Upper classes for the increments of the fractional Wiener process

Summary LetX(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0, ) with stationary increments and varianceEX ² (t)=t ², anda(t) a positive nondecreasing function witha(t)t. We investigate the a.s. asymptotic behaviour of the incrementsI(t, a (t))=max{X{u+a(t))–X(u): 0ut–a(t)} (and some others that are similarly defined) ast.     

Topological spaces and quasi-varieties

We describe Top ^op and Sob ^op as quasi-varieties by means of suitable schizophrenic objects.Research of the first author supported by grants from the NSERC of Canada and the FCAR du Québec. Research of the second author supported by the Topology grant 40% and by the NATO grant CRG 941330.     

Codings of graphs with binary edge labels

LetG(V,E) be a graph. A mappingf:E{0,1} ^m is called a (binary) coding ofG, if the induced mapping , assigns different vectors to the vertices. For the Boolean sum,f is called aB-code, and for the mod 2 sum anM-code. Letm _B (G) resp.m _M (G) be the smallest lengthm for whichB-codes resp.M-codes are possible. Trivially,m _B (G),m _M (G) log₂|V|. Improving results of Z. Tuza we showm _B (G)log₂|V| + 1,m _M (G)log₂|V|+4.     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Propriétés de continuité fine des fonctions coexcessives

Summary We show at first that if (P _t) is a special standard semi-group on a locally compact space E with a countable base and if u is an excessive function, then the semi-group , superharmonic transform of (P _t) by u, is special standard if (and only if) the function u is regular.Then we prove that if two standard semi-groups in duality verifying Kunita-Watanabe's hypotheses are given and if g is a coexcessive function (excessive for the dual semi-group) then: 1) g is Borel measurable, 2) g is a. s. left continuous and has right limits on the sample path of the process 3) g has a.s. right and left limits on the sample path of the process and 4) g is finely continuous except perhaps on a semi-polar set. If we suppose also that the second semi-group is special standard and that the coexcessive function g is regular, then 1) g is a.s. right continuous and has left limits on the sample paths of the process , and 2) g is finely continuous except perhaps on a polar set.     

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space and the Banach algebraF which can be viewed as non-commutative analogues of the Hardy spacesH ² andH respectively.An inner-outer factorization for any element in as well as characterization of invertible elements inF are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS ₁ ,..., S _n of . This enables us to show that every weakly (strongly) closed unital subalgebra of {(S ₁ ,..., S _n ) F } is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.The first author was supported in part by NSF DMS 93-21369 1991Mathematics Subject Classification. Primary 47D25, Secondary 32A35, 47A67.     

A decision theoretical characterization of weak ergodicity

Summary The relation between the ergodic coefficient and deficiency relative to the least informative experiment is investigated. The result is applied to nonhomogeneous Markov chains (NMC's). Our main result can be described as follows: Given an NMC, define the experiments _n ^(j) for n1 consisting in observing the (n+j)-th state of the chain, the j-th state being the unknown parameter. Then the chain is weakly ergodic if and only if for any j, _n ^(j) converges as n (with respect to deficiencies) to the least informative experiment. It is finally shown that in the homogeneous case, the rate of convergence is always exponential.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

Criteria of regularity for norm-sequences

General sufficient conditions are given for a regularity property of norm-sequences. This regularity of the norm-sequence {T ⁿ } _nN makes possible to associate an isometryV with the operatorT in a similar way as it has been known in the power bounded case.Research partially supported by Hungarian NFS Research grant no. T 022920.     

Sequences of m-orthogonal random variables

A sequence of random variables {X_n} is said to be a sequence of m -orthogonal random variables if E X _n ² < for any n and E(X_kX_j) for k–j>m. Here m is a nonnegative integer. One proves a theorem on the law of the iterated logarithm for a sequence ofm-orthogonal random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 198–202, 1982.