Asymptotics for the<i>L<sup>p</sup></i>-deviation of the variance estimator under diffusion

Doukhan, Paul; León, J. B.

doi:10.1051/ps:2004005

Cited by 2 publications

(4 citation statements)

References 15 publications

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“…Although this CLT has been already proved in Soulier [7], in this paper, we will outline an alternative proof of this result, following the procedure used by Doukhan and León [9]. This will be useful to study the more involved case of the deviations in L p .…”

Section: Results For T Fixedmentioning

confidence: 84%

“…Observe that if N = [1/h n ], then we have that: The proof of Lemma 5 is based on the following lemma proved in Doukhan and León [9].…”

Section: Moments For E N (·) P Pmentioning

confidence: 97%

“…The asymptotic properties of the estimator are studied when ε tends to zero and a CLT is obtained. Doukhan and León [9] also consider a smoothed by convolution diffusion process and prove a CLT for the deviation in L p norm, for a variance estimator.…”

Section: Introductionmentioning

confidence: 97%

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Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

Flores

León

2010

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In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L p norm (p ≥ 1) between the variance and this estimator. The method of the proof consists in writing the L p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.

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Section: Results For T Fixedmentioning

confidence: 84%

“…Observe that if N = [1/h n ], then we have that: The proof of Lemma 5 is based on the following lemma proved in Doukhan and León [9].…”

Section: Moments For E N (·) P Pmentioning

confidence: 97%

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Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

Flores

León

2010

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“…For goodness-of-fit testing purposes, one needs to estimate m empirically, for example by the empirical mean . Many solutions are known to fit the limit variance , by a convenient estimate , as suggested in [19] and the references therein. By doing so, we have , while the rate of convergence for is non-parametric, since a typical estimate of is the value at 0 of the spectral density of X at the origin.…”

Section: Test Of Goodness-of-fitmentioning

confidence: 99%

A Dynamic Taylor’s law

Pena

Doukhan²,

Salhi

2022

J. Appl. Probab.

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Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.

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Asymptotics for theL^p-deviation of the variance estimator under diffusion

Abstract: Abstract. We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001)

Cited by 2 publications

References 15 publications

Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

A Dynamic Taylor’s law

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Asymptotics for theLp-deviation of the variance estimator under diffusion

Abstract: Abstract. We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001)

Cited by 2 publications

References 15 publications

Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

A Dynamic Taylor’s law

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Asymptotics for theL^p-deviation of the variance estimator under diffusion