Parametric estimation for Gaussian fields indexed by graphs

In this paper, using spectral theory of Hilbertian operators, we study a special class of Gaussian processes indexed by graphs. We extend Whittle maximum likelihood estimation of the parameters for the corresponding spectral density and show their asymptotic optimality.     

Stochastic processes indexed by hypergroups II

Further results on weakly stationary processes indexed by hypergroups are presented. The concept of translation operators is developed; processes on orbit spaces and double coset spaces are constructed. It is shown that every weakly stationary process indexed by a hypergroupK with centerC contains a maximalK//C-weakly stationary component. New examples forK-weakly stationary processes are continuous estimates of the mean of a weakly stationary process, isotropic random fields, andK-oscillations.     

Empirical processes indexed by smooth functions

In this paper we derive a general invariance principle for empirical processes indexed by smooth functions. The method is applied to prove bounds for the convergence of the empirical distributions which might be useful to verify asymptotic normality of smooth statistical functionals. As one further application we get the convergence of the so-called empirical characteristic function process.     

Stochastic processes indexed by hypergroups. I

For many applications it is desirable to have extensions of the classical theory of weakly stationary processes to certain classes of nonstationary ones. A large family for which Fourier methods still play a major role is the harmonizable class and some related processes. The purpose of this paper is to initiate the study of a class of stochastic processes with a stationarity condition based on the notion of hypergroups.     

Stopping rules and tactics for processes indexed by a directed set

A new notion of tactic for processes indexed by a directed set is introduced. The main theorem, giving conditions under which tactics can be mapped on stopping times on the line, is applied to reduce some optimal stopping problems in the plane to the same problems on the line. In the case of independent random variables, one achieves a nearly complete reduction of the optimal reward problem to the linear case.     

The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {(ν_n(C)/q(P(C))) 1 _{[P(C) λn]}: C }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λ_n → 0.     

STRONG LAWS FOR α-MIXING SEQUENCE PROCESSES INDEXED BY SETS

ＳＴＲＯＮＧＬＡＷＳＦＯＲα－ＭＩＸＩＮＧＳＥＱＵＥＮＣＥＰＲＯＣＥＳＳＥＳＩＮＤＥＸＥＤＢＹＳＥＴＳ￥ＸＵＢＩＮＧＡｂｓｔｒａｃｔ：ＬｅｔＪ＝｛１，２，．．．｝ｄａｎｄｌｅｔ｛Ｘｊ，ｊ∈Ｊ｝ｂｅａｎａ－ｍｉｘｉｎｇｓｅｑｕｅｎｃｅｗｈｉｃｈｉｓｎｏ...     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Laws of the iterated logarithm for partial sum processes indexed by functions

We establish a bounded and a compact law of the iterated logarithm for partial sum processes indexed by classes of functions. We assume a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.     

Functional Erdös-Rényi laws for processes indexed by sets

We consider a process X(A) indexed by some sets A∈A, where A is a collection of sets. We prove a functional form of the Erdös-Rényi laws. The result may be specialized to get back and sometimes improve previous versions of the classical Erdös-Rényi laws.     

Notes on large deviations for branching processes indexed by a Poisson process

Lithuanian Mathematical Journal - Consider a continuous-time process {ZNt}, where {Zn} is a Galton–Watson process with offspring mean m, and {Nt} is a Poisson process independent of {Zn}. It...     

Lattices of processes for graphs

The structure of the lattice of processes of a finite digraph with entries is investigated. Necessary and sufficient conditions for the isomorphism of a given lattice to the lattice of processes of a certain digraph are obtained. Bibliography: 5 titles. Translated by K. V. Shakhbazyan. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 135–153.     

Rates of growth and sample moduli for weighted empirical processes indexed by sets

Summary Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-ervonenkis class C of sets. These inequalities are particularly useful under the assumption P({CC:P(C)<t})»0 as t»0. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulus for the unweighted empirical process, and to study the ratio P _n/P of the empirical measure to the actual measure.Research supported under an NSF Postdoctoral Fellowship grant No. MCS 83-11686, and in part by NSF grant No. DMS-8301807     

A reduction procedure for coloring perfect K4-free graphs

This paper presents an algorithmic proof of the validity of the Strong Perfect Graph Conjecture for graphs whose largest clique is a triangle. The proof leads to an O(n³) algorithm to 3-color such graphs. In the process, a method is presented to contract a perfect graph into a set of smaller perfect graphs that are (K₄-e)-free.     

A procedure to linearize a class of non-linear systems modelled by bond graphs

A procedure is proposed to obtain the linearization of a class of non-linear physical systems using bond graphs. Also, a junction structure of a non-linear bond graph considering linearly dependent and independent state variables is described. From the junction structure of the non-linear bond graph a procedure to build a linearized bond graph is presented. Finally, an example of a Programmable Universal Manipulation Arm (PUMA) manipulator is given.     

Partially exchangeable processes indexed by the vertices of a k-tree constructed via reinforcement

We define a reinforced stochastic process of random variables indexed by the vertices of a k-tree and with values in a Polish space. The work presents a natural extension from an exchangeable to a partially exchangeable setting of previous work done by the authors.     

A computational procedure for the asymptotic analysis of homogeneous semi-Markov processes

A new algorithm for classifying the states of a homogeneous Markov chain having finitely many states is presented, which enables the investigation of the asymptotic behavior of semi-Markov processes in which the Markov chain is embedded. An application of the algorithm to a social security problem is also presented.     

A note on graphs spanned by Eulerian graphs

We show that the problem raised by Boesch, Suffel, and Tindell of determining whether or not a graph is spanned by an Eulerian subgraph is NP-complete. We also note that there does exist a good algorithm for determining if a graph is spanned by a subgraph having positive even degree at every node.