Limit Theorems for Stochastic Processes

Skorokhod, A. V.

doi:10.1137/1101022

Cited by 655 publications

(268 citation statements)

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“…For each n let x 1,n ≤ x 2,n ≤ · · · ≤ x n,n be a sequence of real numbers and denote by X In the case of Theorem A the random variables X j | ≥ ε}] = 0 for any ε > 0, (1.5) since the sum on the left hand side is 0 for n ≥ n 0 (ε) by the uniform asymptotic negligibility condition (1.3). Thus Theorem A is an immediate consequence of the classical functional central limit theorem for sums of independent random variables (see [20]). Theorem B, due to Rosén [18], describes a different situation: if we sample without replacement, the r.v.…”

Section: Introductionmentioning

confidence: 93%

Non-central limit theorems for random selections

Berkes

Horváth

Schauer

2009

Probab. Theory Relat. Fields

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Selection from finite sets is a basic procedure of statistics and the partial sum behavior of selected elements is completely known under the "uniform asymptotic negligibility" condition of central limit theory. The purpose of the present paper is to determine the asymptotic behavior of partial sums when the central limit theorem fails. As an application, we describe the limiting properties of permutation and bootstrap statistics in case of infinite variance.

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Section: Introductionmentioning

confidence: 93%

Non-central limit theorems for random selections

Berkes

Horváth

Schauer

2009

Probab. Theory Relat. Fields

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“…Thus, In analyzing the asymptotic properties of continuous functions of W 1;n and/or W 2;n , it often su¢ces to analyze the properties of the same functions of the independent standard Wiener processes W 1 ; W 2 ; because of the Skorohod (1956), Dudley (1968), and Wichura (1970) representation theorem. See also Gaenssler, P. (1983, p. 83).…”

Section: Are Business Cycles Due To Complex Unit Roots?mentioning

confidence: 99%

Complex Unit Roots and Business Cycles: Are They Real?

Bierens

2001

Econom. Theory

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In this paper the asymptotic properties of ARMA processes with complex-conjugate unit roots in the AR lag polynomial are studied. These processes behave quite di¤erently from regular unit root processes (with a single root equal to 1). In particular, the asymptotic properties of a standardized version of the periodogram for such processes are analyzed, and a nonparametric test of the complex unit root hypothesis against the stationarity hypothesis is derived. This test is applied to the annual change of the monthly number of unemployed in the US, in order to see whether this time series has complex unit roots in the business cycle frequencies.

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“…The Skorohod metric, see, e.g., [4,11,18,26], was introduced in statistics after it had been observed that the L ∞ norm is not adequate for piecewise continuous functions (processes), since it makes the space of such functions nonseparable, i.e., too large. To explain this from a layman's perspective, let us consider for a moment functions f 1/2+1/n :…”

Section: Lemmamentioning

confidence: 99%

The power of adaption for approximating functions with singularities

Plaskota¹,

Wasilkowski²,

Zhao³

2008

Math. Comp.

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Abstract. Consider approximating functions based on a finite number of their samples. We show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions. More specifically, let F 1 r be the class of scalar functions f : [0, T ] → R whose derivatives of order up to r are continuous at any point except for one unknown singular point. We provide an adaptive algorithm A ad n that uses at most n samples of f and whose worst case L p error (1 ≤ p < ∞) with respect to 'reasonable' function classes F 1 r ⊂ F 1 r is proportional to n −r . On the other hand, the worst case error of any nonadaptive algorithm that uses n samples is at best proportional to n −1/p .The restriction to only one singularity is necessary for superiority of adaption in the worst case setting. Fortunately, adaption regains its power in the asymptotic setting even for a very general class F ∞ r consisting of piecewise C r -smooth functions, each having a finite number of singular points. For any f ∈ F ∞ r our adaptive algorithm approximates f with error converging to zero at least as fast as n −r . We also prove that the rate of convergence for nonadaptive methods cannot be better than n −1/p , i.e., is much slower.The results mentioned above do not hold if the errors are measured in the L ∞ norm, since no algorithm produces small L ∞ errors for functions with unknown discontinuities. However, we strongly believe that the L ∞ norm is inappropriate when dealing with singular functions and that the Skorohod metric should be used instead. We show that our adaptive algorithm retains its positive properties when the approximation error is measured in the Skorohod metric. That is, the worst case error with respect to F 1 r equals Θ(n −r ), and the convergence in the asymptotic setting for F ∞ r is n −r . Numerical results confirm the theoretical properties of our algorithms.

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Limit Theorems for Stochastic Processes

Cited by 655 publications

References 1 publication

Non-central limit theorems for random selections

Non-central limit theorems for random selections

Complex Unit Roots and Business Cycles: Are They Real?

The power of adaption for approximating functions with singularities

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