Confidence bands for quantile function under random censorship

Some new confidence bands are established for the quantile function from randomly censored data. The method does not require estimation of the density function. As an application, we construct bands for the quantile function of the length of fractures in the granitic plutons near Lac du Bonnet, Manitoba, where an Underground Research Laboratory is being built for the nuclear waste disposal program in Canada.Geological Survey of Canada Contribution Number 10386.Research partially supported by an NSERC Canada Grant.Research done while at Carieton University, supported by NSERC Canada and EMR Canada Grants of M. Csörg.     

A Strong Approximation Theorem for Quasi-associated Sequences

By combining the Csorgo-Révész quantile transform methods and the Skorohod-Strassen martingale embedding theorem, we prove a strong approximation theorem for quasi-associated random variables with mean zero and finite （2 ＋ δ）th moment under polynomial decay rate. As a consequence, the decay rate for a strong approximation theorem of associated sequences of Yu （1996） is weakened.     

Rate of convergence of transport processes with an application to stochastic differential equations

Summary We obtain a rate of convergence of uniform transport processes to Brownian motion, which we apply to the Wong and Zakai approximation of stochastic integrals.The research of both authors was supported by a NSERC Canada Grant and by an EMR Canada Grant of M. Csörgö at Carleton University, Ottawa     

Strong and weak approximations of k-spacings processes

Summary We obtain strong approximation results for the empirical process and the quantile process based on non-overlapping k-spacings using a characterization of uniform spacings in terms of independent exponential random variables. Moreover the weak convergence of these processes in /q-metrics is discussed. Applications to spacing statistics are included.Part of this research was done while the author was a visiting scientist at the Department of Mathematics and Statistics, Carleton University, supported by Canadian NSERC operating grant of M. Csörg. This research was also supported by a research grant of the Research Center of the College of Science, King Saud University     

A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences

Summary With the help of an extension of Hoeffding's equality, we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables. Based on these estimations, a Glivenko-Cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, all under the conditions on covariances of the original random variables.This research has been supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University, Ottawa, and by an NSERC Canada grant of M. Csörg     

On lattice-isomorphic abelian groups

Research partially supported by grants from Carleton University and N.S.E.R.C. (Canada)     

On a new law of the iterated logarithm of Erdős and Révész

Research supported by the Fok Yingtung Education Foundation and by an NSERC Canada Scientific Exchange Award at Carleton University, Ottawa, Canada.     

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext¹-groups. A stronger condition requiring the surjectivity of the induced maps between Ext ^k -groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras. Research partially supported by NSERC of Canada and by Hungarian National Foundation for Scientific Research grant no. 1903 Research partially supported by NSERC of Canada     

A Sharp Form of the Cramér–Wold Theorem

The Cramér–Wold theorem states that a Borel probability measure P on ℝ ^d is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space. First author partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2002-04430-C02-02. Second author partially supported by Instituto de Cooperación Iberoamericana, Programa de Cooperación Interuniversitaria AL-E 2003. Third author partially supported by grants from NSERC and the Canada research chairs program.     

The Odd-Distance Plane Graph

The vertices of the odd-distance graph are the points of the plane ℝ². Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ² is countably choosable, while such a graph in ℝ³ is not. The research of J. Maňuch was supported in part by MITACS (Mathematics of Information Technology and Complex Systems). The research of M. Rosenfeld was supported in part by the Chancellor Research Grant and the Institute of Technology, UWT. The research of S. Shelah was supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by NSF grant No. NSF-DMS 0600940. No. 923 on Shelah’s publication list. The research of L. Stacho was supported in part by NSERC (Natural Science and Engineering Research Council of Canada) grant.     

Integer-valued functions and increasing unions of first countable spaces

We determine those regular cardinals κ with the property that for each increasing κ-chain of first countable spaces there is a compatible first countable topology on the union of the chain. AssumingV=L any such κ must be weakly compact. It is relatively consistent with a supercompact cardinal that each κ>w ₁ has the property. The proofs exploit the connection with interesting families of integer-valued functions. Research of the second author supported by OTKA grant no. 1805. Research of the remaining authors partially supported by NSERC of Canada.     

Path properties of <Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-valued Gaussian random fields

General limit theorems are established for l ^p -valued Gaussian random fields indexed by a multidimensional parameter, which contain both almost sure moduli of continuity and limits of large increments for the l ^p -valued Gaussian random fields under Ṅ explicit conditions. This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0     

On almost sure limit inferior for B-valued stochastic processes and applications

Summary A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.Work supported by an NSERC Canada grant at Carleton UniversityWork supported by the Fok Yingtung Education Foundation of China     

The dimension of the finite subsets κ

We answer a question of M. Pouzet by showing that the Dushnik-Miller dimension of the finite subsets of the infinite cardinal ordered by inclusion is ().This paper was written when the first author was visiting the University of Calgary, June, 1995. Research partially supported by Office of Naval Research grant NOOO14-90-1206.Research supported by NSERC grant #69-0982.     

On additive functionals of Markov chains

Let{S _n } _n=0 ^∞ be a Harris-recurrent Markov chain on a measurable state space. We prove strong approximation results for the additive functionals . Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. Research supported by an NSERC Canada Grant, Carleton University. Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6.     

The Lévy-Hinčin representation for random compact convex subsets which are infinitely divisible under Minkowski addition

Summary This paper establishes the Lévy-Hinčin representation for all random compact convex subsets of ℝ which are infinitely divisible for Minkowski addition. Research partially supported by NSF grants No. MCS 8100728 and DMS-8318610 Research partially supported by NSF grant No. MCS 8301326     

Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. M. T. Barlow’s research was partially supported by NSERC (Canada), the twenty-first century COE Program in Kyoto University (Japan), and by EPSRC (UK). R. F. Bass’s research was partially supported by NSF Grant DMS-0601783. T. Kumagai’s research was partially supported by the Grant-in-Aid for Scientific Research (B) 18340027 (Japan).     

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups. Research partially supported by an NSERC grant. Alfred P Sloan Fellow. Research partially supported by the NSF.     

Large deviations for Markov processes with mean field interaction and unbounded jumps

Summary An N-particle system with mean field interaction is considered. The large deviation estimates for the empirical distributions as N goes to infinity are obtained under conditions which are satisfied, by many interesting models including the first and the second Schlögl models.Supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University and the NSERC operating grant of D.A. Dawson     

Effective counting of the points of definable sets over finite fields

Given a formula in the language of fields we use Galois stratification to establish an effective algorithm to estimate the number of points over finite fields that satisfy the formula This work was partially done while all three authors were members of the Institute for Advanced Studies in Jerusalem. Parially supported by BSF grant #87-00038 and NSA grant MDA 904-91-H-0057. Partially supported by grants from the German-Israeli Foundation for Scientific Research and Development.