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Local limit theorems for transition densities of Markov chains converging to diffusions
Probability Theory and Related Fields. 2000. Vol. 117. No. 4. P. 551-587.
Konakov V., Mammen E.
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition
densities are proved.
Konakov V., Mammen E., Probability Theory and Related Fields 2009 Vol. 143 No. 1 P. 137-176
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by ...
Added: December 4, 2012
Konakov V., Mammen E., Bernoulli: a journal of mathematical statistics and probability 2005 Vol. 11 No. 4 P. 591-641
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. We prove Edgeworth-type expansions of order o(n-1-δ),δ>0, for transition densities. For this purpose we apply the parametrix method to represent the transition density as a functional of densities of sums of independent and identically distributed variables. Then we apply Edgeworth expansions ...
Added: December 4, 2012
Durmus A., Moulines E., Naumov A. et al., / Cornell University. Series arXiv "math". 2023.
In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein-type inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. Our proof technique is, as far as we know, ...
Added: June 18, 2023
Kelbert M., Sazonov I., Gravenor M., Mathematical Biosciences 2016 Vol. 274 P. 45-57
We consider the epidemic dynamics in stochastically interacting population centers coupled by a random migration. ...
Added: February 15, 2016
Konakov V., Menozzi S., Molchanov S., Annales de l’Institut Henri Poincaré 2010 Vol. 46 No. 4 P. 908-923
For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean-Singer type for the density. We there from derive an explicit Gaussian upper bound and a partial lower bound that characterize the additional singularity ...
Added: December 4, 2012
Konakov V., Kozhina A., Menozzi S., ESAIM: Probability and Statistics 2017 Vol. 21 P. 88-112
We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients. ...
Added: April 14, 2017
Kelbert M., Сухов Ю. М., М. : МЦНМО, 2021
Для освоения теории вероятностей и математической статистики тренировка в решении задач и выработка интуиции важны не меньше, чем
изучение доказательств теорем; большое разнообразие задач по этому предмету затрудняет студентам переход от лекций к экзаменационным задачам,
а от них—к практике. Специфический предмет этого тома, цепи Маркова и их применения,
переживает последнее время большой подъем. Многие замечательные теоретические результаты были ...
Added: February 1, 2021
Konakov V., / Издательство попечительского совета механико-математического факультета МГУ. Серия WP BRP "STI". 2012. № 2012.
В основе настоящего учебного пособия лежит специальный курс по выбору студента, прочитанный автором на механико - математическом факультете МГУ им. М.В. Ломоносова в 2010-2012 учебных годах. Пособие знакомит читателя с методом параметрикса и его дискретным аналогом, развитым в самое последнее время автором пособия и его коллегами-соавторами. Оно объединяет воедино материал, который ранее содержался только в ...
Added: December 5, 2012
Lipatov M., Вестник Московского университета. Серия 1: Математика. Механика 2010 № 5 С. 61-64
The paper proves the recurrence of measurable cocycles with values in the subgroup of SL(2, C) of diagonal and skew-diagonal matrices over an ergodic transformation preserving the probability measure. ...
Added: November 15, 2012
Igor Kheifets, Saikkonen P., Econometric Reviews 2020 Vol. 39 No. 39 P. 407-414
Smooth transition autoregressive models are widely used to capture nonlinearities in univariate and multivariate time series. Existence of stationary solution is typically assumed, implicitly or explicitly. In this paper we describe conditions for stationarity and ergodicity of vector STAR models. The key condition is that the joint spectral radius of certain matrices is below 1, ...
Added: February 23, 2021
Konakov V., Mammen E., / Cornell University. Series arXiv "math". 2023. No. 2304.10673.
The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, Gaussian limit distributions approximate the stochastic performance of the algorithm. Here, we are interested in strong approximations for Robbins-Monro procedures. The main tool for ...
Added: April 24, 2023
Batsyn M. V., Kalyagin V. A., Научно-технический вестник Санкт-Петербургского государственного университета информационных технологий, механики и оптики 2008 № 51 С. 361-372
В работе исследована проблема вычисления функции распределения суммы страховых выплат при наличии эксцедентного перестрахования больших ущербов по индивидуальным страховым контрактам. Получено аналитическое выражение для этой функции при равномерном распределении ущерба в одном страховом случае. Для решения этой задачи использована однопериодная модель коллективных рисков. ...
Added: October 9, 2012
Goryainova E. R., Goryainov V. B., Наука и образование: электронное научно-техническое издание 2011 № 10
Для процесса 2D-авторегрессии порядка (1,1) устанавливается асимптотическая нормальность М-оценок с необязательно выпуклой функцией потерь. При помощи компьютерного моделирования устанавливается устойчивость этих оценок при засорении наблюдений грубыми выбросами. ...
Added: November 29, 2012
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 ...
Added: December 4, 2012
Borodin A., Olshanski G., Journal of Functional Analysis 2012 Vol. 263 P. 248-303
We construct a four-parameter family of Markov processes on infinite Gelfand–Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand–Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(∞). The process has a ...
Added: February 25, 2013
Blank M., Discrete and Continuous Dynamical Systems 2021 Vol. 41 No. 4 P. 1649-1665
We study qualitative properties of the set of recurrent points of
finitely generated free semigroups of measurable maps. In the case of a single
generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but
otherwise it turns out to be possible that almost all points are ...
Added: October 21, 2020
Н. Ю. Энатская, Е. Р. Хакимуллин, М. : Юрайт, 2015
Учебник представляет три вероятностных дисциплины: теорию вероятностей, математическую статистику и случайные про-цессы, составляющие основу вероятностного образования студентов. По каждой дисциплине изложены основные теоретичес-кие вопросы и приведены многочисленные примеры и задачи для иллюстрации теории и пояснения ее практического исполь-зования. Кроме решенных задач по всем главам учебника предложены задачи для самостоятельного решения и теоретические вопросы для самоконтроля ...
Added: November 10, 2015
Olshanski G., Selecta Mathematica, New Series 2021 Vol. 27 Article 41
Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from Ω𝑁 to Ω𝑁−1 for each 𝑁=2,3,…. The elements of the sets Ω𝑁 are the vertices of the extended Gelfand–Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...
Added: June 4, 2021
Belomestny D., Moulines E., Samsonov S., Statistics and Computing 2022 Vol. 32 No. 1 Article 16
In this paper, we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete-time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the ...
Added: August 31, 2020
Bogachev V., Veretennikov A., Shaposhnikov S., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 460 № 5 С. 507-511
Методами уравнений в частных производных установлены достаточные условия дифференцируемости инвариантных мер диффузионных процессов по параметру ...
Added: October 11, 2015
Bezhaeva Z., Куликов В. Л., Олехова Е. Ф. et al., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 38-45
We define an invariant Erdős measure on the compact abelian group of A-adic integers. We also define an A-invariant Erdős measure on the n-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. ...
Added: September 7, 2017
Konakov V., Menozzi S., Electronic Journal of Probability 2017 Vol. 22 P. 1-47
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices. ...
Added: May 12, 2017
Budkov Y., Kolesnikov A. L., Journal of Statistical Mechanics: Theory and Experiment 2016 Vol. 2014 P. 1-12
We propose a new approach based on the path integral formalism to the calculation of the probability distribution functions of quadratic quantities of the Gaussian polymer chain in d-dimensional space, such as the radius of gyration and potential energy in the parabolic well. In both cases we obtain the exact relations for the characteristic function ...
Added: October 27, 2016
Veretennikov A., Теорiя Ймовiрностей та Математична Статистика 2016 Vol. 95 P. 178-188
Poisson equation in the whole space was studied earlier for so called ergodic generators L corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for ...
Added: October 17, 2017