The Small Ball Asymptotics in Hilbert Norm for the Kac--Kiefer--Wolfowitz Processes

Nazarov, Alexander I.; Petrova, Yu. P.

doi:10.1137/s0040585x97t987752

Cited by 8 publications

(6 citation statements)

References 17 publications

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“…For example, if one considers test for normality (with estimated mean and/or variance) then Kac-Kiefer-Wolfowitz processes (KKW processes) appear (see [28]). The exact L 2 -small ball asymptotics for KKW processes is found in [29] and for some other important Durbin's processes in [30].…”

Section: Introductionmentioning

confidence: 89%

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$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

Petrova¹

2019

Preprint

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In this article we study the small ball probabilities in L 2 -norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: non-critical, partially critical and critical; and derive small ball asymptotics for the perturbated process in terms of the small ball asymptotics for the original process.The natural examples of such perturbations appear in statistics in the study of empirical processes with estimated parameters (the so-called Durbin's processes). We show that the Durbin's processes are critical perturbations of the Brownian bridge. Under some additional assumptions, general results can be simplified. As an example we find the exact L 2 -small ball asymptotics for critical perturbations of the Green processes (the processes which covariance function is the Green function of the ordinary differential operator).

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Section: Introductionmentioning

confidence: 89%

“…Lemma 4. Let f (x) : [0, +∞) → R, f ′ (x) is non-decreasing and f (x) ∼ A x α L(x) exp(−Dx −β ), as x → +0, α ∈ R, C > 0, β > 0, D > 0, (29) where L(x) > 0 is a slowly varying function at zero. Then…”

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confidence: 99%

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

Petrova¹

2019

Preprint

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“…This approach was elaborated in [18], [15] and was used in a number of papers, see, e.g., [20] and references therein. We mention also the papers [16], [21], [24] where the twoterm spectral asymptotics was obtained for finite-dimensional perturbations of the Green Gaussian processes.…”

Section: Introductionmentioning

confidence: 99%

Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

Nazarov¹

2019

Preprint

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“…First steps were made in a special case when a Gaussian process is mixed with some finite-dimensional "perturbation". The general theory was built in [18], later some refined results were obtained in the case of Durbin processes (limiting processes for empirical processes with estimated parameters), see [19] as a typical example.…”

Section: Introductionmentioning

confidence: 99%

On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

Nazarov

Nikitin

2018

Mathematics

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We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen-Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.

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The Small Ball Asymptotics in Hilbert Norm for the Kac--Kiefer--Wolfowitz Processes

Cited by 8 publications

References 17 publications

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

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