Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow.
The materials of the 5th International conference on stochastic methods are presented including the following directions: probability and statistics (analytic modelling, asymptotic methods and limit theorems, stochastic analysis, Markov processes and martingales, actuarial and financial mathematics, et al.); applications of stochastic methods (queueing theory and stochastic networks, reliability theory and risk analysis, probability in indistry, economics and other areas, computer science and computer networks, machine learning and data analysis, etc.).
A new notion of ``Markov up'' processes is discussed. A toy model of this notion is proposed and discussed; recurrence and ergodic properties are studied.
We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.
Two--sided bounds are constructed for a probability density function of a weighted sum of chi- square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.
We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result.
We prove that a joint distribution of a locally integrable increasing process X◦ and its compensator A◦ at a terminal moment of time can be realized as a joint terminal distribution of another locally integrable increasing process X* and its compensator A*, A* being continuous.