Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

On the central limit theorem in R k

Summary Let F ^*n denote the n th convolution of a distribution function F on R ^k and suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F ^*n (n·) converges to a Gaussian d.f. as n + t8. These d.f.s determine measures F ^*n (nA) and (A) for Borelsets A, We present a method that admits the estimation of the remainder-term F ^*n (n A)- (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n ^–1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.     

Recurrence and transience for random walks with stationary increments

Summary Let S _n = ₁+...+ _n , n1, be the partial sums of stationary, dependent random variables in ^m . The probability space can be partitioned into I _t I _r , where I _t = {S _n} and I _r ={each S _n is limit point of (S _n)_n1}. This result follows from the inclusion{S _n > for n>0}I _t a.s., which is obtained by using Kac's inequality.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

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.     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.     

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.     

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.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Замечание О Механичецкой Квадратуре

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Asymptotics with respect to a parameter of solutions of linear functional-differential equations

We study the question of the number of linearly independent solutions of the equationy ⁽ⁿ⁾ (x)+(Fy) (x)+ⁿ y (x)=0,x [0, 1], in which F is a bounded linear operator acting on various normed function spaces. A number of assertions about the asymptotic behavior of these solutions with respect to , tending to infinity are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1460–1469, November, 1990.     

Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

It is well known, that for the sums of i.i.d. random variables we have S _n/n 0 a.s. iff _n=1 1/n P(|S _n| > n) < holds for all > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.     

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     

Some properties of the moduli of families of curves

Let A={a₁,...,a_n} and B={b₁,...,b_m} be systems of distinct points in , let be a family of homotopic classes H_i,i=1,..., j+m, of closed Jordan curves on, where the classes H_j+l, l=1,...,m, consist of curves that are homotopic to a point curve in b. Let =₁,..., _j+m be a system of positive numbers and letU be the modulus of the extremal-metric problem for the family and the system . In this paper we investigate the dependence of the modulusU=U(,A,B) on the parameters _i and on the disposition of the points a_k and b. One shows thatU is a smooth function of the indicated arguments and one obtains expressions for the derivatives U, U, and U. One gives some applications of these results.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 72–82, 1985.     

A note on the independent domination number of subset graph

The independent domination number i(G) (independent number (G)) is the minimum (maximum) cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree 1/2n satisfies i(G) 2n/3 1/2. For 1 k l m, the subset graph S _m (k, l) is the bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(S _m (k, l)) and prove that if k + l = m then Havilands conjecture holds for the subset graph S _m (k, l). Furthermore, we give the exact value of (S _m (k, l)).This work was supported by National Natural Sciences Foundation of China (19871036).     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

On weak convergence of functionals of stochastic processes with continuously differentiable paths

We obtain a criterion for weak convergence of a sequence of stochastic processes _n(t), t [0, 1],n N, _n(t) R ^m in the spaceC _m ^k [0, 1] of continuously differentiable functions. We consider several examples of weakly convergent sequences of stochastic processes inC _m ^k [0, 1] and several integer functionals defined on these random variables.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 85–90, 1987.     

On Galois representations arising from towers of coverings of P1\{0, 1, ∞}

Summary We propose a new way to describe, universally, thel-adic Galois representations associated to each almost pro-l tower of etale coverings ofP ¹\{0, 1, }. This generalizes our universal power series for Jacobi sums (cf. [I]) which arises from the tower of Fermat curves of degreel ⁿ (n), and contains the case of the tower of modular curves of level 2ml ⁿ (m: fixed,n) as another important special case. As a fundamental tool, we shall establish and use an almost pro-l version of the theorems of Blanchfield and of Lyndon in Fox free differential calculus.     

Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.