On the chernoff-savage theorem for dependent sequences

Summary Given a sequence of ϕ-mixing random variables not necessarily stationary, a Chernoff-Savage theorem for two-sample linear rank statistics is proved using the Pyke-Shorack [5] approach based on weak convergence properties of empirical processes in an extended metric. This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers. Research partially supported by the National Research Council of Canada under Grant No. A-3954.     

On the strong mixing and weak Bernoulli conditions

Summary A strictly stationary random sequence is constructed which has exactly two states and satisfies the strong mixing condition but not the weak Bernoulli condition.This research was partially supported by the National Science Foundation and the Office of Naval Research     

Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f ² (x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate. Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.     

Fully discrete approximation methods for the estimation of parabolic systems and boundary parameters

A spatially and temporally discrete numerical approximation scheme is developed for the identification of a class of semilinear parabolic systems with unknown boundary parameters. The identification problem is formulated as a least squares fit to data subject to an equivalent representation for the dynamics in the form of an abstract evolution equation. Finite-dimensional difference equation state approximations are constructed using a cubic spline-based, Galerkin method and the Padé rational function approximations to the exponential. A sequence of approximating identification problems result, the solutions of which are shown to exist and, in a certain sense, approximate solutions to the original identification problem. Numerical results for two examples, one involving the modeling of biological mixing in deep sea sediment cores, and the other, the estimation of transport parameters for indoor mixing, are discussed. In both examples, the identification is based upon actual experimental data.Parts of the research were carried out while the authors were visitors at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, Virginia, which is operated under NASA Contracts No. NAS1-17070 and No. NAS1-17130.Research supported in part by NSF Grant MCS-8205355, AFOSR Contract 81-0198 and ARO Contract ARO-DAAG-29-K-0029.     

Eventually independent sequences

The class of eventually independent sequences for a weakly mixing transformation is an isomorphism invariant that is useful for distinguishing zero entropy transformations. This invariant is used to distinguish certain weakly mixing transformations as well as certain partially mixing transformations. In memory of Shlomo Horowitz Research partially supported by National Science Foundation Grant MCS7606735A01.     

Rank one lightly mixing

A rank one transformationT was constructed by Chacón that is weakly mixing but not mixing. We will show thatT is lightly mixing, not partially mixing, and not lightly 2-mixing. Partially supported by an NSF Postdoctoral Research Fellowship.     

不同分布ρ混合序列乘积和的强收敛性

研究了ρ混合序列的收敛性质,利用得到的结果和ρ混合序列的矩不等式讨论了ρ混合序列乘积和的强收敛性质.     

On the convergence of partial differential equations of parabolic type with rapidly oscillating coefficients to stochastic partial differential equations

In this paper we consider the convergence of the solutions to a sequence of partial differential equations of parabolic type with rapidly oscillating coefficients to the solutions of a stochastic partial differential equation. We use the martingale method and the characteristic functional to prove that the martingale problem has a unique solution. Our emphasis is in treating strongly mixing noises.This research was partially supported by Stifting Volkswagenwerk through Forschungszentrum BiBoS Universität Bielefeld and Grant-in-Aid for General Scientific Research (C) 61540162, the Ministry of Education, Science and Culture (Japan). Presented at the International Workshop on Diffusion Approximations and Related Topics, IISA, Laxenburg, Austria, 29 June to 3 July 1987.     

不同分布φ混合序列乘积和的强收敛性

利用φ混合序列矩不等式讨论了φ混合亭列乘积和的强收敛性质．     

Upper Large Deviations for Mixing Random Sequence

In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.     

A class of two-type point processes

Summary A construction is given of a class of two-type point processes with Poisson marginals but possibly negative correlation between points of different types. Examples of the construction are given. The correlation structure of the processes is determined, and criteria obtained for the processes to be stationary, ergodic and mixing.Research supported by Social Science Research Council Spatial Data Project     

Simple Markov‐chain algorithms for generating bipartite graphs and tournaments

We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the near‐regular case, i.e., when all the degrees are almost equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs, which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near‐regular. The proof techniques we use for the two problems are similar. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 293–308, 1999     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

混合序列0-1律及其在完全收敛性中的应用

下面若无特别申明,均设{X_n}为 λ-混合,ρ-混合或φ-混合序列;λ(n),ρ(n),(?)(n)分别为相应的混合速度,它们的定义见[1];均设 C 为绝对常数,即使在同一式中可取不同     

Examples of entropy generating sequence

We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

Ergodic properties where order 4 implies infinite order

For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders. Here we establish this for the properties minimal self-joinings, simplicity and for cartesiandisjointness. An application of the first yields an analog to Kalikow’s celebrated result that for rank-1 transformations, 2-fold mixing implies 3-fold mixing. Via a joining argument we show that for any rank-1 ℤ ^D -action, 4-fold mixing implies mixing of all orders. Indeed, the rank need only be sufficiently close to 1 for the implication to hold and so this result is new even when the acting group is ℤ. By means of limit-joinings, we settle affirmatively an old open question by establishing, for anyM, thatM-fold Rényi-mixing impliesM-fold mixing. Partially supported by National Science Foundation Postdoctoral Research Fellowship.     

Maximal Inequalities and an Invariance Principle for a Class of Weakly Dependent Random Variables

The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle for strongly mixing sequences of random variables in the absence of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences.     

Banach空间上的强混合算子

1999年,L．B．Gonzalez 证明了任意无限维可分 Banach 空间上存在拓扑传递的有界线性算子．这个结果肯定地回答了 S．Rolewicz 提出的问题．本文证明了由 L．B．Gonzalez 所给出的算子实际上是强混合的,同时,对加权移位算子的混合性利用权序列进行了刻划并指出任意无限维可分 Hilbert 空间上存在弱混合而非强混合的有界线性算子．