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Of all publications in the section: 3
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Article
Ulyanov V. V., Goetze F., A. Naumov. Journal of Theoretical Probability. 2017. Vol. 30. No. 3. P. 876-897.

In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.

Added: Mar 17, 2016
Article
Naumov A., Tikhomirov A., Гётце Ф. Journal of Theoretical Probability. 2019. P. 1-36.

We consider a random symmetric matrix $$\X = [X_{jk}]_{j,k=1}^n$$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $$\max_{jk} \E |X_{jk}|^{4+\delta} < \infty, \delta > 0$$, it was proved in \cite{GotzeNauTikh2016a} that with high probability the typical distance between the Stieltjes transforms $$m_n(z)$$, $$z = u + i v$$, of the empirical spectral distribution (ESD) and the Stieltjes transforms  $$m_{sc}(z)$$ of the semicircle law is of order $$(nv)^{-1} \log n$$.  The aim of this paper is to remove $$\delta>0$$ and show that this result still holds if we assume that $$\max_{jk} \E |X_{jk}|^{4} < \infty$$. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

Added: Apr 25, 2019
Article
Konakov V., Menozzi S. Journal of Theoretical Probability. 2011. Vol. 24. No. 2. P. 454-478.

Consider a multidimensional stochastic differential equation governed by a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to a time step for the difference of these densities.

Added: Dec 4, 2012