Probability Theory

Chow, Y. S.; Teicher, Henry

doi:10.1007/978-1-4684-0504-0

Cited by 357 publications

(108 citation statements)

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“…Central limit theorems of type (44) has a substantial history. The classical Lindeberg-Feller (cf Section 9.1 in Chow and Teicher (1988)) concerns independent random variables. Hoeffding and Robbins (1948) proved a central limit theorem under m-dependence.…”

Section: Central Limit Theorymentioning

confidence: 99%

Asymptotic theory for stationary processes

2011

Statistics and Its Interface

105

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We present a systematic asymptotic theory for statistics of stationary time series. In particular, we consider properties of sample means, sample covariance functions, covariance matrix estimates, periodograms, spectral density estimates, U -statistics, kernel density and regression estimates of linear and nonlinear processes. The asymptotic theory is built upon physical and predictive dependence measures, a new measure of dependence which is based on nonlinear system theory. Our dependence measures are particularly useful for dealing with complicated statistics of time series such as eigenvalues of sample covariance matrices and maximum deviations of nonparametric curve estimates.

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Section: Central Limit Theorymentioning

confidence: 99%

Asymptotic theory for stationary processes

2011

Statistics and Its Interface

105

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“…By Doob's martingale convergence theorem (see, e.g., [25] or [26], or any textbook on advanced probability theory), it therefore converges on a set of -measure one. Moreover, converges on a set of measure one, being a positive super-martingale as well [ Thus, must converge on a set of measure one.…”

Section: Proofmentioning

confidence: 99%

“…Another possible choice is the Hellinger distance (25) and (26) Like the square distance, the Hellinger distance is bounded by both the relative entropy and the absolute distance and (27)…”

Section: Static MDLmentioning

confidence: 99%

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Asymptotics of Discrete MDL for Online Prediction

Poland

Hutter

2005

IEEE Trans. Inform. Theory

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Abstract-Minimum description length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning processes which are independent and identically distributed (i.i.d.) by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e., observations come in one by one, and the predictor is allowed to update its state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely, a static and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the Kullback-Leibler loss of the MDL learner, which are, however, exponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely, sequence prediction, pattern classification, regression, and universal induction in the sense of algorithmic information theory among others.

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“…In this paper, we shall deal with random non-self-maps. First, by employing a result of Chow and Teicher [6], sufficient conditions are obtained for a random non-self-map, so that the existence of a deterministic fixed point is equivalent to the existence of a random fixed point. Thus every fixed point theorem produces a random fixed point theorem.…”

Section: Introductionmentioning

confidence: 99%

On Deterministic and Random Fixed Points

Tan¹,

Yuan²

1993

Proceedings of the American Mathematical Society

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Abstract.Based on an extension of Aumann's measurable selection theorem due to Leese, it is shown that each fixed point theorem for F{w, •) produces a random fixed point theorem for F provided the rj-algebra X for Sl is a Suslin family and F has a measurable graph (in particular, when F is random continuous with closed values and A' is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

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Probability Theory

Cited by 357 publications

References 0 publications

Asymptotic theory for stationary processes

Asymptotic theory for stationary processes

Asymptotics of Discrete MDL for Online Prediction

On Deterministic and Random Fixed Points

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