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The dyadic fractional diffusion kernel as a central limit

Hugo Aimar, Ivana Gómez, Federico Morana (2019)

Czechoslovak Mathematical Journal

We obtain the fundamental solution kernel of dyadic diffusions in ℝ + as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution of classical Fourier analysis by Haar wavelet analysis.

The empirical distribution function for dependent variables: asymptotic and nonasymptotic results in 𝕃 p

Jérôme Dedecker, Florence Merlevède (2007)

ESAIM: Probability and Statistics

Considering the centered empirical distribution function Fn-F as a variable in 𝕃 p ( μ ) , we derive non asymptotic upper bounds for the deviation of the 𝕃 p ( μ ) -norms of Fn-F as well as central limit theorems for the empirical process indexed by the elements of generalized Sobolev balls. These results are valid for a large class of dependent sequences, including non-mixing processes and some dynamical systems.

The first exit of almost strongly recurrent semi-Markov processes

Joachim Domsta, Franciszek Grabski (1995)

Applicationes Mathematicae

Let ( · ) , n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding renewal kernels on K×K to J×J are assumed to be suitably convergent to a renewal kernel P (on J×J). The HSMP on J corresponding to P is assumed to be strongly recurrent. Let [ π j ; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged...

The law of the iterated logarithm for the multivariate kernel mode estimator

Abdelkader Mokkadem, Mariane Pelletier (2003)

ESAIM: Probability and Statistics

Let θ be the mode of a probability density and θ n its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θ n - θ . Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θ n - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the l p norms, p ∈ [ 1 , ∞ ] , of θ n - θ ....

The law of the iterated logarithm for the multivariate kernel mode estimator

Abdelkader Mokkadem, Mariane Pelletier (2010)

ESAIM: Probability and Statistics

Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the...

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Loïc Hervé, Françoise Pène (2010)

Bulletin de la Société Mathématique de France

The Nagaev-Guivarc’h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order Edgeworth expansion,...

The rate of convergence for spectra of GUE and LUE matrix ensembles

Friedrich Götze, Alexander Tikhomirov (2005)

Open Mathematics

We obtain optimal bounds of order O(n −1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model

Yuliya Mishura (2015)

Banach Center Publications

We take the martingale central limit theorem that was established, together with the rate of convergence, by Liptser and Shiryaev, and adapt it to the multiplicative scheme of financial markets with discrete time that converge to the standard Black-Scholes model. The rate of convergence of put and call option prices is shown to be bounded by n - 1 / 8 . To improve the rate of convergence, we suppose that the increments are independent and identically distributed (but without binomial or similar restrictions...

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