Error bounds on multivariate Normal approximations for word count statistics

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

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Error bounds on multivariate Normal approximations for word count statistics

Advances in Applied Probability ◽

10.1239/aap/1033662166 ◽

2002 ◽

Vol 34 (3) ◽

pp. 559-586 ◽

Cited By ~ 4

Author(s):

Haiyan Huang

Keyword(s):

Covariance Matrix ◽

Error Bounds ◽

Multivariate Distribution ◽

Finite Alphabet ◽

Multivariate Normal ◽

Word Count ◽

First Order ◽

Normal Approximations ◽

Necessary And Sufficient ◽

The Given

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w1), …, N(S,wd)), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O((log n) / √n) in the i.i.d. case and O((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

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Resolution systems and their applications I

Fundamenta Informaticae ◽

10.3233/fi-1980-3209 ◽

1980 ◽

Vol 3 (2) ◽

pp. 235-268

Author(s):

Ewa Orłowska

Keyword(s):

Theorem Proving ◽

Sufficient Conditions ◽

Predicate Calculus ◽

Resolution Method ◽

First Order ◽

Deductive Systems ◽

Resolution System ◽

Resolution Principle ◽

Necessary And Sufficient ◽

Classical Predicate

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science. In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic L in another logic L ′ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic L and some sets of formulas of the logic L ′ with the intention of using the resolution system for L ′ to prove theorems of L. It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.

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Necessary and sufficient conditions for oscillation of first-order nonlinear neutral difference equations

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2007.06.009 ◽

2008 ◽

Vol 55 (6) ◽

pp. 1279-1292 ◽

Cited By ~ 2

Author(s):

X.H. Tang ◽

Xiaoyan Lin

Keyword(s):

Difference Equations ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Neutral Difference Equations ◽

First Order ◽

Necessary And Sufficient

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On Minimum-B Stabilization of Electrostatic Drift Instabilities

Zeitschrift für Naturforschung A ◽

10.1515/zna-1966-1118 ◽

1966 ◽

Vol 21 (11) ◽

pp. 1953-1959 ◽

Cited By ~ 2

Author(s):

R. Saison ◽

H. K. Wimmel

Keyword(s):

Dispersion Relation ◽

Particle Distribution ◽

Inhomogeneous Plasma ◽

Distribution Functions ◽

Loss Cone ◽

First Order ◽

Stabilization Theorem ◽

The Stability ◽

Necessary And Sufficient ◽

Limiting Case

A check is made of a stabilization theorem of ROSENBLUTH and KRALL (Phys. Fluids 8, 1004 [1965]) according to which an inhomogeneous plasma in a minimum-B field (β ≪ 1) should be stable with respect to electrostatic drift instabilities when the particle distribution functions satisfy a condition given by TAYLOR, i. e. when f0 = f(W, μ) and ∂f/∂W < 0 Although the dispersion relation of ROSENBLUTH and KRALL is confirmed to first order in the gyroradii and in ε ≡ d ln B/dx z the stabilization theorem is refuted, as also is the validity of the stability criterion used by ROSEN-BLUTH and KRALL, ⟨j·E⟩ ≧ 0 for all real ω. In the case ωpi ≫ | Ωi | equilibria are given which satisfy the condition of TAYLOR and are nevertheless unstable. For instability it is necessary to have a non-monotonic ν ⊥ distribution; the instabilities involved are thus loss-cone unstable drift waves. In the spatially homogeneous limiting case the instability persists as a pure loss cone instability with Re[ω] =0. A necessary and sufficient condition for stability is D (ω =∞, k,…) ≦ k2 for all k, the dispersion relation being written in the form D (ω, k, K,...) = k2+K2. In the case ωpi ≪ | Ωi | adherence to the condition given by TAYLOR guarantees stability.

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System Properties of One-Dimensional Distributed Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3427094 ◽

1977 ◽

Vol 99 (2) ◽

pp. 85-90 ◽

Cited By ~ 4

Author(s):

L. S. Bonderson

Keyword(s):

Distributed Systems ◽

Sufficient Conditions ◽

State Variables ◽

One Dimensional ◽

First Order ◽

Lumped Element ◽

Order Form ◽

Power Product ◽

System Properties ◽

Necessary And Sufficient

The system properties of passivity, losslessness, and reciprocity are defined and their necessary and sufficient conditions are derived for a class of linear one-dimensional multipower distributed systems. The utilization of power product pairs as state variables and the representation of the dynamics in first-order form allows results completely analogous to those for lumped-element systems.

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Bipartite graphs with close domination and k-domination numbers

Open Mathematics ◽

10.1515/math-2020-0047 ◽

2020 ◽

Vol 18 (1) ◽

pp. 873-885

Author(s):

Gülnaz Boruzanlı Ekinci ◽

Csilla Bujtás

Keyword(s):

Positive Integer ◽

Dominating Set ◽

Domination Number ◽

Bipartite Graphs ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Minimum Cardinality ◽

Vertex Set ◽

Necessary And Sufficient ◽

The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

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A General Analysis of Sequential Social Learning

Mathematics of Operations Research ◽

10.1287/moor.2020.1093 ◽

2021 ◽

Author(s):

Itai Arieli ◽

Manuel Mueller-Frank

Keyword(s):

Social Learning ◽

Utility Function ◽

Posterior Probability ◽

General Analysis ◽

Sufficient Condition ◽

Important Class ◽

General Utility ◽

Learning Game ◽

Necessary And Sufficient ◽

The Given

This paper analyzes a sequential social learning game with a general utility function, state, and action space. We show that asymptotic learning holds for every utility function if and only if signals are totally unbounded, that is, the support of the private posterior probability of every event contains both zero and one. For the case of finitely many actions, we provide a sufficient condition for asymptotic learning depending on the given utility function. Finally, we establish that for the important class of simple utility functions with finitely many actions and states, pairwise unbounded signals, which generally are a strictly weaker notion than unbounded signals, are necessary and sufficient for asymptotic learning.

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Existence of nonoscillatory solutions of first order nonlinear neutral equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000008419 ◽

1990 ◽

Vol 32 (2) ◽

pp. 180-192 ◽

Cited By ~ 6

Author(s):

Lu Wudu

Keyword(s):

Nonoscillatory Solution ◽

Sufficient Condition ◽

Neutral Equation ◽

Necessary And Sufficient Condition ◽

Nonoscillatory Solutions ◽

Neutral Equations ◽

First Order ◽

Image Position ◽

Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

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On B- Covariant Derivative of First Order for Some Tensors in different Spaces

Journal of Mathematical Analysis and Modeling ◽

10.48185/jmam.v2i2.276 ◽

2021 ◽

Vol 2 (2) ◽

pp. 30-37

Author(s):

Alaa A. Abdallah ◽

A. A. Navlekar ◽

Kirtiwant P. Ghadle

Keyword(s):

Curvature Tensor ◽

Covariant Derivative ◽

Torsion Tensor ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

First Order ◽

Recurrence Property ◽

Necessary And Sufficient ◽

The Relationship

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

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