Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

On the postulation of a general projection of a curve inP 3

Summary We say that a curve C in P ³ has maximal rank if for every integer k the restriction map r_c(k):H ⁰(P ³, O_P3(k)) H⁰ (C, O_C(k))has maximal rank. Here we prove the following results. Theorem 1Fix integers g, d with 0g3,dg+3.Fix a curve X of genus g and L Pic^d (X).If g=3and X is hyperelliptic, assume d8. Let _L(X)be the image of X by the complete linear system H ⁰(X, L). Then a general projection of _L(X)into P ³ has maximal rank. Theorem 2For every integer g0,there exists an integer d(g, 3)such that for every dd(g, 3),for every smooth curve X of genus g and every LPic^d (X) the general projection of _L(X)into P ³ has maximal rank.     

On the Skitovich–Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f _n: XX a homomorphism f _n(x)=nx, and X ⁽ⁿ⁾=Im f _n. Denote by I(X) the set of idempotent distributions on X and by (X) the set of Gaussian distributions on X. Consider linear statistics L ₁= ₁( ₁)+ ₂( ₂) and L ₂= ₁( ₁)+ ₂( ₂), where _j are independent random variables taking on values in X and with distributions _j, and _j, _jAut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L ₁ and L ₂ implies that ₁, ₂I(X) if and only if X possesses the property: for each prime p the factor-group X/X ^(p) is finite. If X is connected, then there exist independent random variables _j taking on values in X and with distributions _j, and _j, _jAut(X) such that L ₁ and L ₂ are independent, whereas ₁, ₂(X) * I(X).     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

This paper is devoted to a study of the properties of the equationA ^*FA–F=–G, where FL() is unknown, AL(), GL() is positive and is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemx _k+1=Ax _k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rⁿ is considered.This work was supported in part by the Polish Academy of Sciences under the contract Problem Miedzyresortowy I.1, Grupa Tematyczna 3 This paper was written while the author was with the Instytut Automatyki, the same university.     

A partial ordering of some of the elements of Algol

Them _Algol productions of Algol are of the formX _p0 ::=X _p1 X _p2 ...X _{pn p} , where 1pm _Algol, 1n _p,X _p0 is a defined type andX _pj, 1jn _p is either a defined type or a basic symbol or possibly, representing the empty string ifn _p=1 [1]. A partial ordering of that subset of Algol's basic symbols and defined types which for somep are eitherX _p0 orX _p1 is exhibited. This ordering is of interest in implementing the syntax-oriented translator described by Ingerman.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

Extremal problems in classes of univalent functions,omitting prescribed values

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f()=c₁+c₂²+... regular and univalent in the circleU={||<1} and such that f(₁)f(₂)=1 for ₁₂U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f(₀)| in the class of functions f()R with a fixed value f(₀, where ₀ is an arbitrary point U, and of the maximum of |f(₀)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.     

Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus)

Let X ⁴ be a smooth hypersurface of degree d 5, and let S X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z Pic(X) of degree 0, whose image in CH₁(X) is in the kernel of the Abel–Jacobi map. The family of couples (X, S) containing such Z is a countable union of analytic varieties. We show that it has a unique component of maximal dimension, which is exaclty the locus of couples (X, S) satisfying the following condition: There exists a line S and a plane P ⁴ such that P X = , and Z = – dh, where h is the class of the hyperplane section in CH₁(S). The image of Z in CH₁(X) is thus 0. This construction provides evidence for a conjecture by Nori which predicts that the Abel–Jacobi map for 1–cycles on X is injective.     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen

Let X=X ₁,...,X _n be the ring of formal power series inn indeterminates over . LetF:XAX+B(X)=(F ⁽¹⁾(X),...,F ⁽ⁿ⁾(X))(X) ⁿ denote an automorphism of X and let ₁,..., _n be the eigenvalues of the linear partA ofF. We will say thatF has an analytic iteration (a. i.) if there exists a family (F _t (itX)) _t of automorphisms such thatF _t(X) has coefficients analytic int and such thatF ₀=X,F ₁=F,F _t+t=F_tF_t for allt,t. Let now a set=(ln₁,...,ln _n ) of determinations of the logarithms be given. We ask if there exists an a. i. ofF such that the eigenvalues of the linear partA(t) ofF _t(X) are . We will give necessary and sufficient conditions forF to have such an a. i., namely thatF is conjugate to a semicanonical formN=T ^–1FT such that inN ^(k)(X) there appear at most monomialsX ₁ ¹ ...X _n ⁿ . This generalizes a result of Shl.Sternberg.

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Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

Commutator Subgroups of the Extended Hecke Groups \bar H(\lambda _q )

Hecke groups H(_q) are the discrete subgroups of generated by S(z) = –(z+ _q)^–1and T(z) = –1/z. The commutator subgroup of H(_q), denoted by H(_q), is studied in [2]. It was shown that H(_q) is a free group of rank q– 1.Here the extended Hecke groups obtained by adjoining to the generators of H(_q) are considered. The commutator subgroup of is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H(_q) case, the index of H(_q) is changed by q, in the case of this number is either 4 for qodd or 8 for qeven.     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

On the existence of solutions of operator differential equations

We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L _p(0, T; W_p¹()L₂ (0, T; L₂()) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 837–850, June, 2004.     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

Calculi with monotone deductions and their economic interpretation

Let the possible ways of development of some system from the initial state X₀ be given by the deductive system <;X₀> (X₀ is an axiom, the algorithm defines the relation of deducibility in one step). Let Y₁,..., Y_e be all states directly derivable from X [i.e., (X)={y₁,..., Y_e]. Let be an algorithm assigning for each X transition probabilities p₁,..., p _l , where is the transition probability to the special state STOP. defines a probability measure on the set of all deductions. We define the information in the pair < ;X₀> by the forla: where px is the probability of being in X directly before STOP. We consider , assigning a fixed p for each X and satisfying the condition p₁= ...=p _l . Then the information in < ; X₀> becomes a function <;X₀> of one p. The essential characteristic of the system <;X > is given by the asymptotic behavior of <;X₀> as p 0. This characteristic corresponds well with the intuitive notion of the relative power of calculi. Now we consider <,X>(p) as a function of X. For many types of systems there is a useful strategy for maximizing this function (the strategy of increasing freedom of choice); we consider in this connection the simplest systems of economic character. Let X, Y, Z be n-dimensional vectors with nonnegative components (the components are interpreted as resources and products of a certain economic system, gives the technological possibility of transformations of the resources). Let Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 90–104, 1979.